The 23-year-old, who favours a central striking role, is frustrated at being used regularly in other positions.
He has two years left on his contract but is debating whether to move rather than accept a new deal if offered one.
Asked about Welbeck's future after United's 2-0 defeat at Everton on Sunday, manager David Moyes said: "Danny Welbeck is really important to me and I really value him."
Moyes added: "As you know, there have been a lot of stories."
Welbeck scored more goals for England than United last season, scoring three times in seven matches for Roy Hodgson's side (strike rate: 0.43 goals per game), compared to twice in 40 appearances under then Reds boss Sir Alex Ferguson (strike rate: 0.05).
Welbeck has scored 10 times in 32 United appearances this season and is expected to be in Roy Hodgson's England squad for this summer's World Cup.
With Wayne Rooney, Robin van Persie, Javier Hernandez and £37.1m January signing Juan Mata all competing with Welbeck for a starting position up front for United, manager David Moyes is unlikely to offer him any assurances about next season.
There would be no shortage of suitors should he opt to leave - although United know that the Manchester-born academy graduate is held in affection by the club's supporters.
Welbeck scored only two goals last season, although one of those came against Real Madrid at the Bernabeu.

Cherryl Henry-Leach from Doncaster Children's Services Trust, which runs Getting On, said it was one of the only programmes of its kind in the UK.
Families work with police, the health service, voluntary sector and council to avoid violence at home.
Ms Henry-Leach said parents were often reluctant to get help.
"They feel shame and stigma attached to being a victim of domestic abuse, while also fearing the consequences for their child if reported," she said.
The programme seeks to stop children moving from one abusive relationship to another, using a "whole-family approach" unique to Doncaster to build families' self-esteem and resilience, Ms Henry-Leach said.
Many teenagers on the programme have lived with domestic violence between parents so the project attempts to "de-normalise" abuse and help them move on to healthier relationships.
Mother and son, Anne and Luke (not their real names) from Doncaster, said the techniques derived from University of Oxford research taught them how to listen.
"There was a lot of swearing and abusive behaviour before," Anne said.  "Luke never actually hit me, but he'd go to hit me, or push me which could be frightening.
"I can now sit down and talk to my son without us shouting at each other. We can actually have a conversation without being abusive.
Luke agreed that they have a "healthier relationship."
Five Doncaster families have now completed the nine week programme, which is funded by Â£3.1m from the Department for Education's Innovation Fund.
A further six families are still on the programme.
Manchester United striker Danny Welbeck is considering leaving the club.
[NEXT_CONCEPT]
A project aiming to tackle violence suffered by parents at the hands of their children has been hailed a success in Doncaster.