The England international goalkeeper spent the 2016-17 campaign on loan at Italian club Torino.
The 30-year-old does not seem to be in City manager Pep Guardiola’s side plans.
Butland, who has been on the books of Stoke since January 2013 when he joined from Birmingham City for a transfer fee reported by The Daily Mail to be worth £4 million, has spoken highly of Hart.
“He had a tough time at the beginning of last season at Manchester City when he wasn’t part of their plans,” Butland told The Daily Star about Hart.
“The type of character he is, he went to Italy to play football and take himself away from everything. I hope to one day be in his position and I wouldn’t want be on the receiving end of that.
“We get on really well. We both have the same steely determination, nothing fazes him and he just gets on with it. He doesn’t care what people say about him. I’ve learnt that from him from a very young age.”
The 30-year-old central midfielder is a good player, but the Englishman has not been at his best so far this season.
West Ham have struggled in the opening weeks of the 2017-18 Premier League campaign and so has Noble.
Former West Ham winger Sinclair has defended Noble, and has told BBC Sport: "I've heard people say he is finished, but they have got short memories. I know Mark personally - I know him well - and there is still plenty of football left in him.”