Kicker says Bayern are ready to sell Costa and Renato Sanches, which will put United on red alert.
And Italian sources say Costa will make his mind up about potentially moving to Old Trafford if United secure Champions League football.
Victory in the Europa League final tonight against Ajax will see United return to the elite competition.
Tuttosport reports United are not alone in the race to sign Costa with Juventus also interested in securing his signature.
Costa made it clear that he was not happy at Bayern earlier this season.
“I don't completely feel happy here, but we will soon find a solution. Whether I stay it depends on a few factors," Costa said.
“Main factor is minutes played. Being a core player is a crucial difference in my career.
“England is one of the biggest leagues in the world – fantastic."