England left-back Baines was reportedly the subject of a £12million bid from United in midweek, but the offer is understood to have been swiftly rejected by the Toffees.
Despite claims from Goodison Park that Baines is not for sale, David Moyes is expected to try again to lure the 28-year-old defender down the East Lancs Road.
Barkley, who is currently out in Turkey with the England Under-20 squad at the U20 World Cup, has spoken of his respect for Baines and his strong desire for him to stay on Merseyside.
"I've been following what is happening at Everton and I desperately want Leighton to stay," said Barkley.
"He's someone I look up to, a role model in how he conducts himself and the strides he has made in his career."
Barkley will be in action for England U20s on Saturday night in their third and final Group E clash against Egypt in Bursa (kick-off 7pm BST).
Having drawn against Iraq and Chile, Peter Taylor's men will be going all out for victory as they look to secure a last-16 berth.
However, it has been over 16 years since England last won a game at this tournament, when Michael Owen netted the only goal against Mexico in Johor Bahru, Malaysia, at the 1997 World Youth Championship.
This is the fifth U20 tournament England have competed in since then, and they have now gone 16 games in total without a triumph.
Pressure is on
"There is pressure on us, but we are confident," Barkley wrote in his column in the Daily Express.
"The spirit is good and our training and preparation is very thorough.
"Peter Taylor, our coach, and his assistants Chris Ramsey and Ugo Ehiogu have been doing a lot of work on keeping the ball and counter-attacking swiftly.
"The fact the match kicks off at 9pm local time - 7pm at home- will benefit us because conditions will be more like what we are used to.
"Egypt have lost both their games and I expect them to try to put us under pressure early on, which makes it all the more important for us to control possession."
England U20s have been accused of wastefulness and Barkley admits their failure to finish off opponents has cost them thus far.
"No one here is ready to come home," he added.
"We have to look at ourselves as players as to the reasons why we are in this position and make sure we learn and improve our performances.
"We should have beaten Iraq and in the last half-hour against Chile, we created enough chances to win.
"The brutal truth is that the whole team hasn't been clinical enough in front of goal."