Southampton and Crystal Palace are said to be 'ready for a transfer tug-of-war' over the Portuguese defender.
Both clubs are said to have watched Almeida in action for Benfica - who share Palace's Eagles nickname - in Sunday's Portuguese Super Cup win over Vitoria Guimaraes.
Benfica's Andre Almeida in action against Guimaraes
And it is claimed the Liga NOS champions are 'braced for bids' after the 26-year-old was named man of the match.
The Sun reports Almeida is 'stubbornly resisting' Benfica's attempts to tie him to a new contract after setting his sights on a move to the Premier League, and is valued at £8 million.
Nevertheless, it remains to be seen whether Southampton or Crystal Palace will look to pursue their alleged interest in the player, despite the tabloid's claim a 'transfer tug-of-war' is set to ensure between the two clubs.
Southampton already have Jeremy Pied and Cedric Soares competing in Almeida's position, while Palace are expected to sign versatile Manchester United defender Timothy Fosu-Mensah this week.
Prospective Palace recruit Timothy Fosu-Mensah in pre-season action for Manchester United
Having said that, Soares has been tentatively linked with a move away from St Mary's, while Fosu-Mensah could be used at centre-back at Selhurst Park.
Almeida is also capable of filling multiple roles, playing at left-back and defensive midfield last season.
Could Southampton / Crystal Palace move for Almeida?
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