Transport Secretary Grant Shapps said the UK was "very open to hosting the final round" of the competition, which is due to take place between English clubs Manchester City and Chelsea in Istanbul on May 29.
The English Football Association were in talks with European football's governing body UEFA already, he said, but added the choice of venue for the final was a decision for UEFA.
Shapps said Chelsea and City supporters should not travel to Istanbul, telling a Downing Street press conference: "I'm afraid we are having to put Turkey on the red list and this will have a number of ramifications.
"First of all, it does mean with the regards to the Champions League, fans should not travel to Turkey.
"The FA are in discussions with UEFA already on this.
"We are very open to hosting the final but it is ultimately a decision for UEFA.
"Of course ... the UK has already got a successful track record of football matches with spectators, so we are well-placed to do it."
Shapps added: "We are very open to it ... it is actually, in the end, a decision for UEFA to make, but given there are two English clubs in that final, we look forward to hearing what they have to say."
UEFA Champions League