The claim has certainly been made on a number of occasions. In an article titled Chained to scourge of slavery in the Sydney Morning Herald (dated 6 December 2012), the Australian journalist, Elizabeth Farrelly, observed:
The United Nations estimates there are more slaves in the world now than ever. Human trafficking - which is not the same as slavery though the two are clearly linked, since most slaves are trafficked and most trafficking ends in slavery - rates with arms and drug trafficking among the world's richest illicit industries.
As regards definitions, "slavery" is defined in article 1 of the Slavery Convention of 1926, as:
“the status or condition of a person over whom any or all of the powers attaching to the right of ownership are exercised”.
There is no internationally agreed definition of servitude, but the term is generally used to describe a condition of serfdom, without implying an element of ownership of the victim as the term “slavery” does.
It isn't clear exactly which report is being referred to in the article quoted above (if only journalists cited their sources!). There are a few possibilities. A strong contender is the Global Report on Trafficking in Persons, 2012.
So, in short, it seems that the answer is yes. It appears that there are actually now more slaves in the world than ever before.
Obviously, it must be noted that the global population is greater now, but the statistic is nevertheless particularly remarkable since slavery is illegal in most modern countries today.