Smalling is out of contract at Old Trafford at the end of the season and would be available on a free transfer.
The centre-back has seen improved form in recent weeks following some disappointing performances at the start of the campaign.
It came after a summer in which Manchester United manager Jose Mourinho was hunting for a new central defender.
Smalling is believed to have already held talks over a new deal at Old Trafford.
The Sun claim that he wants improved wages on his current £90,000-a-week deal.
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Regardless, the same outlet also report that Everton are hoping to lure Smalling away on a free transfer.
It’s claimed that Everton boss Marco Silva has sent scouts to watch Smalling a number of times.
A spanner could be put in Everton’s plans with United having the option of extending Smalling’s deal for a further 12 months.
It’s claimed that wouldn’t put Everton off and could even mean a deal is done sooner.
United could be tempted by an offer of £30million for the England defender, regardless of when his contract ends.
A United source told the Sun: “Chris has been looked at by a number of clubs, but Everton have shown a lot of interest.
“He is out of contract next summer and there’s been chats over extending it, but nothing has been finalised as of yet.
“There needs to be some protection of the investment in him, so don’t be surprised if the new year deal comes in.
“If that doesn’t happen then he will walk away at the end of the season. He can also talk to European clubs in January, we’re aware of that.”
Smalling has made ten Premier League appearances for United this season but only helped the Red Devils keep one clean sheet.
Manchester United sit eighth in the table after 12 games, 12 points behind leaders Manchester City.