Spurs are also interested and have increased their own opening offer.
Pierre-Emile Hojbjerg is set to join Everton in a £25m deal Credit: PA:Press Association
But it is understood to be well short of the deal on the table from the Toffees.
Hojbjerg, 25, is free to leave Saints after refusing to sign a new contract.
The ex-Bayern Munich ace was stripped of the captaincy for doing so during lockdown.
Ralph Hasenhuttl wanted a player committed to the cause to skipper his side – and gave the armband to James Ward-Prowse.
And with only one year remaining on Hojbjerg’s current terms, the Austrian boss is ready to cash in to use the funds elsewhere.
Saints are waiting for the green light on a work permit for Ghanaian defender Mohammed Salisu of Valladolid in a £10.9m deal.
Full-backs are also in demand with a deal to turn Kyle Walker-Peters’ loan from Tottenham likely.
While Hojbjerg’s sale could also see the South-Coast side move for a new central midfielder this summer.