Straw made the comments during Parliament questions, prompted by famed games critic and Labour MP Keith Vaz.
Following the banning of controversial title Manhunt 2 and the furore over PC game Law and Order: Double or Nothing, which featured an image of murdered toddler James Bulger, Vaz said: "There is a clear need for better regulation of the videogames industry."
"Will the Leader of the House please tell us when he expects a statement to be made, or when we may have a debate on the social responsibilities of those who make a huge amount of money out of these videogames?" Vaz added.
Straw, responding to the comments made by Vaz, said: "We do not see sufficient social responsibility and understanding by the creators and purveyors of such games.
"I will of course ensure that the [Prime] Minister is made fully aware of my right honourable Friend's concerns."