A TV insider said: "He has been offered the part and we are waiting to hear if he will accept. He is the perfect choice.”
Kinnear has been a favourite for the role since Smith announced his departure earlier this month, and has previously played MI6 officer Bill Tanner in James Bond films 'Skyfall' and 'Quantum of Solace'.
Jenna-Louise Coleman is preparing to say goodbye to 'her' Doctor - Matt Smith
However, showrunner Steven Moffat has insisted that a new Doctor has not yet been found, after it was rumoured an announcement was due to be made this weekend.
TV producer and director Ed Stradling, tweeted that there is going to be a while to wait until audiences find out who has landed the part.
He tweeted: "Don't believe what you read about the new Doctor Who being announced this weekend. I think it must be Starburst getting a bit over-excited!
"Steven Moffat says this morning 'I haven't a clue who it is, we've barely started.' So no Doctor Who announcement this evening I fear!"