Conservative MP Rob Wilson has secured an urgent question to the culture secretary over whether she believes the corporation has gone far enough to deal with the claims.
According to BBC News, Wilson also intends to ask why the corporation dropped a Newsnight investigation into Savile at the end of last year.
The BBC is launching two inquiries into the claims of sexual abuse against Savile, who presented Top of the Pops and Jim'll Fix It for the BBC in the 1970s and 1980s, as well as appearing on Radio 1 for many years.
He died in October 2011, aged 84, but a year later there have been numerous allegations against him.
Copyright: flickr/BBC PicturesPolice believe that the TV star may have sexually abused 60 people since 1959, and are currently following up more than 350 lines of inquiry.
One of the BBCs investigations will look at the "culture and practices of the BBC during the years Jimmy Savile worked here".
The second, being led by BBC Scotland director Ken MacQuarrie, will look into why the Newsnight investigation was dropped.
Wilson is expected to express his concern over how the BBC has dealt with the scandal in his question to be asked later today.
Meanwhile, former Radio 1 DJ Dave Lee Travis has denied claims that he groped two women while at BBC Studios.
One woman, said to have been 17 at the time of the alleged incident, told the Daily Mail that Travis put his hand up her skirt in 1977, while another women alleged that he touched her breasts when she appeared on Radio 4 in the 1980s.
However, Travis said in a statement on Sunday: "I categorically deny that there is any substance in either allegation and I'm genuinely surprised that allegations of this nature have been made. I totally refute any impropriety."