Rashford has nine goals in his last 10 games for club and country - the best run of his career - but the 22-year-old has said there is more to come.
Pochettino was the red-hot favourite to replace Jose Mourinho when he was sacked by United previous year, but Solskajer was appointed as interim manager before landing the job full-time.
Five wins from their last six matches before the global break helped lift the mood at Old Trafford somewhat though, with Rashford's goals helping United get back on track.
British media reported that Rodgers was Spurs chairman Daniel Levy's top candidate while Leipzig's Julian Nagelsmann was also considered before the North London club chose to entrust Mourinho with the task of reviving their fortunes.
Tottenham Hotspur's Kyle Walker-Peters, Mauricio Pochettino, Ben Davies and Danny Rose celebrate after the UEFA Champions League quarter final second leg match at the Etihad Stadium, Manchester last season on their way to the final.
Rashford admits Solskjaer played a key role in him signing his new contract in July, as he can see a clear plan for the club.
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"I told them that one of the reasons why I made a decision to come was them".
There are no sporting objections for Alderweireld and Vertonghen with Mourinho's preferred style of football and the report also mentions the manager tried to sign the former when at Manchester United.
"If we managed to win a trophy and get top four it would be a successful season".
"To me, it's just about breaking the markers for what we can do in the future because we know this season won't change overnight, Ole won't get what he wants overnight".
"We were on the same page and wanted the same things for the club".
Rashford stated: "Ole is a good man and he has the membership's pursuits at coronary heart".
"We have to harden our time and just be patient, as long as we don't continue to try and believe in what we do - I think the team will do 100 percent - then I think we'll be fine".