Pogba admitted in June that "it could be a good time to have a new challenge somewhere else", while agent Mino Raiola has also twice publicly spoken about a possible transfer. Similarly, Lukaku's representative Federico Pastorello has also spoken about moving on.
But when Solskjaer faced the media for the first time this season from United's training base in Australia, he denied either star is close to leaving in what has been dubbed a 'strong message'.
"As far as I'm aware there have been no bids for any players. We are Man United and we don't have to sell. It's business as usual," the Norwegian is quoted as saying by the Daily Mail.
"Paul has never been a concern for me. He's working hard, he's a true professional and a proud boy, same answer again. I can't be sitting here talking about what Paul and his agent are saying. He's got a few years left of his contract and he's been fantastic.
"Agents talk all the time and we've not had any bids, that's all I can say for this matter.
Raiola's latest involvement in the saga saw the agent defend Pogba over his apparent desire to leave the club, insisting the Frenchman has done nothing wrong and remains 'respectful and professional'. He did, however, also bizarrely attack United for not addressing the criticism directed at Pogba following the player's public acknowledgment of his desire to leave the club.
The Sun reports that, privately, Solskjaer, who is described as 'desperate' in the story, will ask Pogba to stay at Old Trafford for one more year and help drive the rebuilding project.
But a separate report from the tabloid newspaper claims the reasons that a 'disaffected' Pogba wants to leave is because he doesn't see United being in a position to win major trophies.
Real Madrid are the clear favourites to land the 2018 World Cup winner, although Marca appear somewhat pessimistic after reporting that United remain determined not to sell and that it could take a club record offer of €180m (£162m) to convince them otherwise.
As far as Lukaku is concerned, it has been heavily reported that Inter want the player but are struggling to raise the funds for a suitable offer. That being said, the Serie A side are seemingly not put off, with Inter officials reportedly due in England this week for talks.
The Sun alleges that Solskjaer has asked Lukaku to stay, suggesting the pair spoke before the squad flew to Australia on Sunday and the player was told that he would have chances to play.