The side Pogba joined, Juventus, are also said to be looking at Chong ahead of making a potential move for the 19-year-old winger.
Gomes, who is considered one of the best prospects in the United academy, is reportedly wanted by Barcelona after impressing in his brief moments on the pitch for the first-team.
Having made just one appearance in the Premier League this season, the playmaker has found himself in the under 23 squad more often than not, but has still only made five appearances.
But Solskjaer is adamant that he wants both of the youngsters to stay and stake their claim for a place in the senior team.
Angel Gomes has played in the Premier League 2 more often this season (Image: GETTY)
He opened up on talks between the two parties and seemed confident that a deal can be arranged.
“We've had lots of conversations with the boys," Solskjaer said.
"Obviously their contracts run out in the summer but we want them to stay.
“I can't see a better place for a young lad that's come through the ranks in the academy to play first team football.
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Tahith Chong is wanted by Juventus (Image: GETTY)
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“Both Angel and Chongy have been here for many, many years.
"We just want them to continue their development and hopefully they can see that when young lads take their chance, they'll get more opportunities."
United take on Bournemouth today at the Vitality Stadium [12.30pm].