The United striker will now "assess" United's intent on pushing forward this summer before committing to discussions about his future.
Solskjaer took over as caretaker United boss in December following the sacking of Jose Mourinho and the club went on a run of 11 games without defeat, of which they won 10.
Solskjaer's side will have to make do with Europa League football next term after they missed out on one of the Champions League spots.
The Red Devils are expected to adopt a different approach in the transfer market under Solskjaer by buying young players to develop rather than marquee signings and the United manager apparently views Dembele as the ideal candidate to lead the line of a newly formed setup.
However, according to The Sun, Rashford has informed the club he does not want to enter negotiations due to his concerns over the club's ambitions.
Lyon signed Dembele a year ago and the France Under-21 star has scored 20 goals this season, with 14 in Ligue 1 - he was also given games in the Champions League.
Solskjaer, as has been the case in recent weeks, did not name any individuals on his hit list but identified some of the character traits he would prefer in anyone who will move to Old Trafford over the next few months.
And Solskajer wants a young striker in place of the Belgian and Dembele looks like an optimistic option.
Paul Pogba is pictured in the kit but it is unknown as to whether or not he will even be at the club next season given his disappointing displays this season as well as the fans' apparent indifference towards him. While the youngster's family and Monaco both await the decision of the proceedings, the rumour mill is abuzz with reports of Manchester United submitting a huge offer to the midfielder.
And the Brazilian isn't the only superstar on their radar.
"#RedDevils are also interested in Philippe #Coutinho (with the arrival of Griezmann he can leave from Barcellona) and Youri #Tielemans".
"Yesterday new contacts between Borussia Dortmund".