Mourinho told BBC Radio Five Live that he would welcome Gerrard.
"The only thing I can say is if he comes I will welcome him with open arms, if he doesn't come, I will admire him exactly the same. I wish him all the best even if he stays where he is at Liverpool."
Kenyon said: "Steven Gerrard is recognised as one of the best midfielders around and were it possible we would be extremely interested. It's great to see how Chelsea midfielder Frank Lampard and Steven have played together and Steven certainly comes into the Lampard class.
"Wayne Rooney's been the talk of the tournament. He's been incredible, particularly coming off a fairly average season. But there's so much speculation, and if we started reacting to all the speculation we would literally do nothing else."
One player that appears to be off the Chelsea shopping list is Deco.
"Deco is a good player, but we've got Joe Cole and Joe has to feature I think," Kenyon told Chelsea TV.
"I spoke to Joe just before he went off to the Euro's. He was burning to get on and make his mark. He clearly wants to make his mark with Chelsea, and we really want to give him the opportunity. I think Joe Cole again comes back to representing that young English player who's got real quality and we've got to give him that platform."
Kenyon revealed that Chelsea have not yet given up hope of signing Didier Drogba, Marseille have rejected one bid already but Kenyon says Chelsea are keeping tabs on the player.
"I think Drogba has been a phenomenon in the last eighteen months to two years, you know he is big. I think he's twenty-six. He is big and physical, and fast, and you have to think that those are good traits for the Premier League, but again, we've got some top quality strikers and what we've got to do all along is balance what we've got with who else is on the market, and that's something again we're looking at right now."