Boyle used to be on BBC comedy panel show Mock the Week, but left in 2009.
Shane Allen, who moved to the BBC from Channel 4, said: "I'm not going to suddenly stop admiring his unique comic talent because I've switched teams.
"I worked with him at Channel 4 and I admired him, but it wouldn't be my first move to get him on air.
"I want to make sure we have the right vehicle and the right channel and that Frankie would get all the support from the top down.
"The thing about Frankie is he has gone through a couple of media storms."
He added: "For me, Frankie shares the same provocative, edgy comedy danger that Billy Connolly had early in his career and Billy is now hailed as a national treasure. I think TV can accommodate someone like Frankie and it's a shame we didn't have more of Billy Connolly on telly in the 1970s."
Mr Allen said he is also a fan of Mrs Brown's Boys, the comedy that was first a hit at the Pavilion Theatre in Glasgow. He said: "It was a game-changer. I think sometimes people in TV land make TV for people in TV land, and Mrs Brown's Boys is a perfect example of how to serve an audience."
A BBC spokeswoman added: "Shane's comment about Frankie was purely speculative. He was asked about Frankie and was very clear there are no current plans for Frankie on the BBC, but if the right project came up and with the right guidance and support, he could return to the BBC. After all, Frankie was never banned from the BBC."
Boyle caused controversy over comments he made about Katie Price's disabled son and also jokes on Twitter about the Queen and Rebecca Adlington.
His recent performance at a Comic Relief concert was edited out before it was broadcast.