Some of them offer the opportunity to "upgrade" free accounts through paid-for membership, providing more access than with the free account. Others encourage purchases to speed up game play or to give access to extra features.
In particular, the OFT will consider whether these apps push children to buy something or to pester their parents or other adults for it.
"We are concerned that children and their parents could be subject to unfair pressure to purchase when they are playing games they thought were free, but which can actually run up substantial costs," said Cavendish Elithorn, OFT Senior Director for Goods and Consumer, adding that the OFT is not seeking to ban in-game purchases.
The OFT will look into whether the full cost of some of these games is made clear when they are first downloaded.
The BBC reported the case of five-year-old schoolboy Danny Kitchen, from Bristol, who managed to rack up charges of more than 1,700 pounds last month while playing the Zombies versus Ninjas game on his parents' iPad. The money has since been refunded by Apple.
In January, regulator PhonePayPlus said it had seen a fourfold increase in complaints related to children and mobile apps and warned parents of the potential costs of initially free applications.
The OFT said it was unable to say which companies it was investigating and that no assumption should be made that any of them have broken the law.