The Clarets centre-back, and former Red Devil, has repeatedly been linked with a return to Old Trafford ahead of the transfer window.
United struggled for defensive depth as the last campaign wore on, with Daley Blind forced into central defence on a number of occasions.
And Mourinho doesn't want the same situation to develop next year.
The Daily Mail say the Portuguese is putting club chairman under pressure to sign Keane from Burnley.
They say United are willing to offer £24m for the England centre-back, of which they'll keep £6m thanks to a 25 per cent sell-on clause they inserted in his contract when he left for Burnley.
Apparently Mourinho wants the deal over the line as quickly as possible after watching rivals Manchester City snap up Monaco midfielder Bernardo Silva for £43m on Friday night.
But Keane might be more difficult to prise away from Turf Moor than expected, with Clarets chairman Mike Garlick claiming his side would rather turn down transfer offers for Keane this season and let him walk away for free next summer when his contract runs out if it means Burnley staying in the Premier League for another year.
“Let’s roll it forward a year, what’s the prize for staying up again? Another £120-130m?" he said.
“So if we miss out on one or two fees and get that, I know what I’d take.
“The most important thing for the fans to know is that we don’t actually have to sell any player.
“Going forward it’s got to be on our terms and terms that are right for us. Do we need to sell anyone? No. If we want to keep players, we’ll keep them.”
