Ramos was understood to be unsettled earlier in the season after reports emerged of a bust-up with president Florentino Perez after they were knocked out of the Champions League by Ajax.
The Real legend is said to have told Perez to sell him and it appears he might still be keen to leave.
Ramos still has two years left to run on his contract and would command a sizeable fee despite now being 33 years old.
However, he would be a massive coup for whoever could get him and Spanish publication AS claim Liverpool and United have contacted his agent for talks.
United are desperate for a new centre-back and have been linked with almost every big name in Europe.
They do not come much bigger than four-time European champion Ramos who could come into a struggling side and make an immediate impact.
Whether United’s absence from next season’s Champions League would affect his decision remains to be seen but one thing they could offer is the promise of an enormous financial package.
They were keen to sign Ramos before he signed a new deal with Real in 2015 and the player was accused of leading them on the secure improved terms at the Bernabeu.
However, that is unlikely to deter them in revisiting the deal given their desperate need for a world-class centre-half.
United boss Ole Gunnar Solskjaer is desperate to overhaul his squad and bringing in Ramos would go some way to appeasing disgruntled fans.
As well as a new centre-back, the Norwegian is eyeing reinforcements in almost every position.
Liverpool, though, may well have the upper hand as they can guarantee Ramos Champions League football.
And they will begin next season as European champions if they beat Tottenham in this year’s final.
Reds boss Jurgen Klopp has warned fans not to expect another summer of big spending.
But the chance to sign Ramos and pair with him with Virgil van Dijk could be too good to turn down.
Paris Saint-Germain are also said to be interested should Ramos become available