United bosses are keen to appoint a British boss should the club have no choice but to sack Ole Gunnar Solskjaer.
And Old Trafford chiefs believe Rodgers would be keen on taking the job, provided a deal can be struck with his club Leicester.
Rodgers would be reluctant to leave the Foxes, but knows there is little more he can realistically achieve there - and would find the chance to manage United almost impossible to turn down.
Rodgers is believed to have a release clause in his contract should a Champions League club come calling.
But United wouldn't hesitate in meeting it if they need to appoint a new boss.