The Spain international is out of contract at the end of the season, and is yet to agree a new Real Madrid contract.
Carlo Ancelotti has indicated that the club wants to keep him at the club, calling him a "vital" player, but he could be tempted by a return to the Premier League.
Chelsea and United are reportedly preparing rival contract offers to present to Alonso in the January transfer window.
From the New Year, interested clubs will be able to offer Alonso a contract that would see him leave Real Madrid at the expiration of his current deal in the summer.
Real Madrid have already confirmed they will not sell him in January, which means the only way he'll leave the Bernabeu is via a free transfer.
Reports earlier this month revealed Manchester United's interest in the 31-year-old midfielder, and the Daily Star claimed United officials had already held discussions over a potential contract offer.
Likewise, Jose Mourinho is monitoring developments, and would jump at the chance to bring his former-player to Stamford Bridge.
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