Both of the Red Devils stars have scored three times since the interim boss took over from Jose Mourinho in December.
But Solskjaer held talks with Rashford this week and made it clear that he will lead the line, according to the Daily Star.
The England international failed to impress under the management of Mourinho and he was tipped to leave in January.
But the 21-year-old has looked back to his best in recent weeks following goals against Newcastle, Bournemouth and Cardiff.
Speaking about Rashford after the 2-0 win over Newcastle, Solskjaer said: "He gets kicked, he gets tackled, he goes for headers, he challenges, he runs.
"He is going to be a top number nine, definitely."
Lukaku was on compassionate leave when Solskjaer took charge – which gave Rashford the opportunity to impress.
The Belgian striker, who joined from Everton for £75m, has came off the bench to score both of the club's last two Premier League games.
And he scored United's second goal after he started the FA Cup win over Reading.
Solskjaer has urged his players to show off their aggressive side in a bid to up the intensity of training session.
But both of the stars appeared to be in high spirits during the club's warm-weather training camp in Dubai this week.
Rashford even took the chance to poke fun at the Norwegian boss with a hilarious tweet.
The Reds, who have won five on the bounce, will face Tottenham at Wembley on Sunday.
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