Leicester manager Rodgers is favourite to replace the 48-year-old with ESPN reporting United chiefs are keen to recruit from within the Premier League unless they see a dramatic improvement in results.
But according to ESPN, the Red Devils are keen on bringing former Celtic and Liverpool manager Rodgers to Old Trafford.
The report suggests Rodgers has impressed with his work at the King Power Stadium, where he won the FA Cup with the Foxes at the end of last season.
Koeman was sacked after a poor run of form in which they went 1-3 in their last four league games.
Like Barça, United are also having a hard time of it as of late. They scraped a 3-2 win over Atalanta in the Champions League, before losing 5-0 at home to Liverpool in their last game.
Speaking after the annihilation, Solskjaer admitted his future is at stake, but is confident he can turn things around.
"Well, the only way I know how to approach life," he said.
"That's to wake up tomorrow morning, look forward, evaluate what's gone on, what's gone wrong, do my very, very best to come through a difficult time.
"This is the worst feeling. You feel rock bottom and the feeling we've had, that we are getting to where we want to be, has been there.
"But in the last few weeks we've hit a brick wall. The results haven't come, the performances haven't come, we've conceded too many goals, too many easy goals and that's a concern.
"Well, I've heard nothing else [on my future] and I'm still thinking about tomorrow's work.
"Of course, we're all low, I can't say now that I've felt any worse than this, this is the worse I've been, the lowest I've been, but I accept the responsibility and that is mine today and it's mine going forward.
"I do believe in myself, I do believe that I am getting close to what I want with the club.
"I think what we've done, what I've seen, the development, of course the results lately haven't been good enough, it's hands up and that brings doubt in anyone's mind probably.
"But I've got to keep strong and I do believe in what we've been doing, the coaching staff, the players. Today's rock bottom, low in confidence.
"Next week is Tottenham away, it's a Champions League game Atalanta, the next team who visits us is Man City.
"We have to look forward, sort our frames of mind out and make sure we go into the next training session in the right frame of mind."
Man United Fan Terry EXPLODES as Solskjaer ISN'T SACKED! Watch the latest on the Football Terrace...News Now - Sport News