Short Answer
There is no solid evidence that Bamako was a settlement of any importance during the period of the Mali Empire (c. mid 13th century to early 17th century). While there is substantial literary and / or archaeological evidence for the importance of centres such as Timbuktu, Gao, Djenné, Oualata and Niani, the same cannot be said for Bamako. However, this does not mean that there was no occupation at all: it is likely that there were some settlements in the region of Bamako during at least part the period of the Mali Empire, but the continuous settlement of Bamako itself most likely dates back no further than the early 17th century.
The New Encyclopedia of Africa (2nd Ed) entry you mentioned cites only Perinbam as its source, but it misrepresents what she states in her book.
Details
Oral tradition dates settlement of the site to the very tail end of the Mali Empire or even a little after its final demise. None of the Arabic sources (Ibn Khaldun, Ibn Battuta, Leo Africanus) covering the Mali Empire mention a settlement which has been identified as Bamako.
Nous savons...d'après la carte de Ptolémée (IIe siècle après
Jésus-Christ), que de nombreuses localités bordaient dans cette région
la vallée du Niger (Nigir). Mais il ne semble pas avoir existé
d'agglomération importante sur le site même de Bamako. Cuphé ou Koufé
devait se situer à 60 km en amont et la «métropole du Nigir» était
sans doute voisine de Koulikoro. La présence dans ce secteur d'un
centre important s'explique par le fait que le commerce s'effectuait
surtout en utilisant la vallée du Sénégal. Lorsque cette dernière sera
abandonnée, la capitale se déplacera vers le Nord d'où venait le
commerce transsaharien. De là la fortune de Tombouctou. Cette
modification des courants d'échanges explique sans doute qu'aucune
grande ville importante ne soit mentionnée dans la région de Bamako au
temps des grands empires soudanais.
Source: Marie-Louise Villien-Rossi, 'Bamako, capitale du Mali'. In 'Les Cahiers d'Outre-Mer' (1963)
Translation:
We...know from Ptolemy's map (2nd century AD) that many localities in this region bordered the Niger valley (Nigir). But there does not seem to have been any significant agglomeration on the very site of Bamako. Cuphé or Koufé must have been located 60 km upstream and the "metropolis of Nigir" was probably close to Koulikoro. The presence in this sector of an important center is explained by the fact that the trade was carried out especially by using the valley of Senegal. When the latter was abandoned, the capital moved to the North where the trans-Saharan trade came from. Hence the fortune of Timbuktu. This modification of trade flows probably explains why no large major city is mentioned in the Bamako region in the time of the great Sudanese empires.
We cannot be certain when Bamako was founded as we have only oral traditions to go on. Of these, the Niare settler tradition is most prominent.
Les traditions orales de Bamako sont claires. Les Niaré, des Soninké
originaires de Nioro, sont les fondateurs du village. La tradition des
Maures Tawati (ou Touré) parle de l'antériorité du passage d'un
certain Bemba, mais elle précise que celui-ci avait quitté les lieux
avant l'arrivée de l'ancêtre des Niaré.
Selon la généalogie des Niaré, la création du village se situerait
vers 1640 (Meillassoux, 1963).
Source: Balla Diarra, Moise Ballo & Jacques Champuaud, 'Structure urbaine et dynamique spatiale à Bamako' (2003)
Translation: The oral traditions of Bamako are clear. The Niaré, Soninké from Nioro, were the founders of the village. The tradition of the Moors Tawati (or Touré) speaks of the anteriority of the passage of a certain Bemba, but it specifies that this one had left the place before the arrival of the ancestors of the Niare.
According to the genealogy of the Niaré, the creation of the village dates back to around 1640 (Meillassoux, 1963).
So, there is some evidence here of prior settlement if the Tawati tradition is correct. B. Marie Perinbam, who has made a detailed study of the oral traditions relating to Bamako, notes the obvious problems with all these stories: there is an obvious temptation to adjust the narrative to suit various interested parties. Thus, she writes:
Whether or not settlers — in addition to the numuw — were already in the
Bamako region prior to the Niare arrival remains problematic, although
it is highly likely that riverside camps for transient Bozo and/or
Somono families were already in place. Oral accounts, moreover, speak
of hunters in the Minkungu region (on the Manding Plateau overlooking
the Bamako plain), who may have been associated with Niare lineages.
An all-tooslender discourse in the archaeological literature
substantiates this with claims of fifteen archaeological sites in or
near the Bamako region, some Neolithic, possibly related to hunters,
traders, and/or to initiation ceremonies,...
Source: B. Marie Perinbam, 'Family Identity and the State in the Bamako Kafu, c. 1800—c. 1900' (1997)
Perinbam mentions archaeology but makes little of it. Further investigation of this led to an article by Kléna Sanogo & Nafogo Coulibaly, titled 'La problématique des « tumulus pierriers » au Mali' (trans: The problem of “stone mounds” in Mali). This mentions that there are a number sites within the boundaries of modern Bamako, but there are no dates and the sources it cites do not seem to relate to these sites and no further details are forthcoming. However, there are details on sites in the Bamako region and beyond, and these are also indicated on the map below (source).
Finds such as these in the Méma region of Mali might go a long way towards providing evidence for earlier (and even continuous) occupation of the Bamako site but there doesn't appear to be anything. Possibly, evidence has been destroyed by subsequent urban development, and important artefacts may also have been lost to the illegal trade in antiquities. Thus, we cannot assume that absence of evidence is, in itself, evidence. However, there really is nothing to support the assertion made in the New Encyclopedia of Africa, vol. 1 that
...the settlement [Bamako] grew to become a prominent center of
Islamic learning during Mali’s medieval period (twelfth to fourteenth
centuries).
As Perinbam's book is the only source cited at the end of the article, and as Perinbam does not make any such claim in her book, the above statement is unsourced and unsubstantiated.
Other sources consulted include:
Claude Meillassoux, 'The Social Structure of Modern Bamako'. In 'Journal of the International African Institute' Vol. 35, No. 2 (Apr., 1965)
D. T. Niane (ed), 'General History of Africa, vol IV: Africa from the Twelfth to the Sixteenth Century' (UNESCO, 1984)
Pascal James Imperato, ' Historical Dictionary of Mali' (3rd ed., 1996)
M. Ruthven, A. Nanji, 'Historical Atlas of the Islamic World' (2004)
Will Fenstermaker and Isabella Garces, 'Excavating Empires of the Sahel: In Conversation with Daouda Keïta' (2020)
