The spinoff's previous host Adrian Chiles left the BBC last month after it emerged that Evans will host his other programme The One Show on Friday evenings.
According to The Sun, the BBC is now considering hiring Evans for You're Fired! as well.
"If Evans is good enough for the Friday One Show he could be good enough for You're Fired! and it would be a cheeky two fingers to Chiles," a source explained. "If Chris was appointed you can imagine how Chiles would feel, with two of his most beloved BBC shows going to his arch rival."
Other candidates for the job reportedly include Sandi Toksvig, George Lamb, and Evan Davis.
Radio 4 business presenter Adam Shaw, newsreader Sophie Raworth and BBC economics editor Stephanie Flanders are also thought to be in the frame.
"It's a big show for BBC Two," the channel's controller Janice Hadlow said. "We want somebody with a big presence to hold a show like that together and someone who is quite a big figure."
> Party leaders pay tribute to Chiles