Yes, there are numerous sources pointing to Bulgarian people in this region long before the officially accepted 681 year. You can check Saint Jerome's map who lived in 4th century. On the map there is a label - Mesia hec & vulgares which is roughly translated as Moesia here also Bulgaria.
The map is a copy from 12th century which is used as an excuse to claim that this was added later. Note that the authenticity of the copy is not questioned and during the 12th century Bulgarian lands are under Byzantine rule and Bulgaria does not exists as a country.
But even without this map, there are plenty of other Latin and Greek sources mentioning Bulgarians living on the Balkans before 7th century. Here you can check some of these sources.
You can also check this genetic study from 2013 which brings forth some very interesting questions about who exactly are the ancestors of modern day Bulgarians.
And because people tend to ignore external links, I will add some citations:
Marcellinus Comes: The Chronicle of Marcellinus
(A. C. 530.) Ind. VIII, Lampadio et Oreste coss. Mundo Illyricianae utriusque
militiae ductor, dudum Getis Illyricum discursantibus primus omnium Romanorum ducum
incubuit, eosque haud paucis ipsorum interemptis fugavit.
His autem deinde coss. idem dux audaciae suae secundus in Thraciam quoque
advolans, praedantes eam Bulgares feliciore pugna cecidit, quingentis eorum
in praelio trucidatis
This can be roughly translated as :
(year 530) Mundo, the master of the Illyrian soldiery, was the first Roman
general to set upon the Goths who had previously been traversing Illyricum
and put them to flight, after quite a few of them had been killed. However,
later in this consulship, this same leader, fortunate in his boldness,
hastened into Thrace and also killed, by fighting bravely, the Bulgars who
were plundering it; five hundred of them were slain in the battle.
or this one from The Chronicles again:
(499) VII. Ioannis Gibbi solius
1 Aristus Illyricianae ductor militae cum quindecim milibus armatorum et cum
quingentis uiginti plaustris armis ad proeliandum necessariis oneratis contra
Bulgares Thraciam deuastantes profectus est. Bellum iuxta Tzurtam fluuium
consertum, ubi plus quam quattuor millia nostrorum aut in fuga aut in
praecipitio ripae fluminis interempta sunt.
This can be translated as
(year 499)Aristus, the leader of Illyria went with 15k soldiers and the
appropriate weaponry against the Bulgarians who were devastating Thrace.
The battle took place at Tzurtam river where more than 4k of our soldiers
died either while running or jumping from the high river bank.
And it is not only Marcellinus Comes, there are also Anastasius Bibliothecarius, Jordanes, John Malalas and others. And while their records do not show firm Bulgarian establishment it is quite clear that Bulgarians have frequented Thrace long before 681.
So obviously I cannot agree with other answers that the date when Bulgarians came on their present day state is well established. It is not. Also we cannot talk about Slavs during 5-8 century. Present day people living on this land never called themselves Slavs. This term appeared much later (16-17 century) to describe all people using Cyrillic alphabet and of course they do have something in common, but this is an entirely another discussion.