The Denmark international is out of contract next June and may therefore be available for a reduced fee before the English transfer window closes on Thursday.
United had been tracking Bruno Fernandes for much of the summer, but the Portugal international has told Sporting CP that he would prefer a move to Tottenham.
And the Red Devils are now set to turn their attention to Eriksen as they seek to bolster their midfield options ahead of the 2019/20 campaign.
Previously a transfer target for Real Madrid, the Danish playmaker had looked set to see out the final year of his contract at Spurs.
However, United have almost agreed personal terms with Eriksen and will now work on agreeing a fee with his current employers ahead of Sunday's clash with Chelsea.
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