The Pittodrie side have already thrown out a £2 million bid from Hull City, and are braced for Swansea City to make a move of their own.
But according to reports, Aberdeen bosses are hopeful of securing a big fee for McKenna, which would go a long way to paying for the club’s new stadium development.
A £10 million price tag may signal the end of Hull’s interest but relegated Swansea could have the money for such a bid.
Swans boss Graham Potter is expected to run the rule over the Scotland international in next week’s Europa League tie between the Dons and Burnley, but has been warned to aim high with any prospective offers.
Former Pittodrie kid Ryan Fraser left the north east for just £400,000 in 2014 and is now a regular English Premier League player with Bournemouth, while highly rated youngsters including Fraser Fyvie and Clark Robertson both left for nothing.
Aberdeen managed to persuade McKenna to sign a new five-year deal keeping him at Aberdeen until 2023; the length of contract practically guaranteeing a sizeable offer if clubs are serious about securing his services.
McKenna was called up to the Scotland squad for the friendly matches agianst Costa Rica, Hungary, Peru and Mexico, and captained the side against El Tri.
According to reports, the Dons believe McKenna will develop into a top player, and have pointed to the meteoric rise of Harry Maguire, who joined Leicester City from Hull last summer in a move that could bank the Tigers up to £17 million.
Hull have been linked with McKenna since February but could be forced to look elsewhere with Aberdeen unlikely to accept a low bid.