"I'm sure if [Arsenal manager] Arsene Wenger wanted a player from me I would have got a call from him but I didn't receive a call," Grant said, as quoted by BBC Sport. "He is our player. He is very committed to the club."
The Hammers, who face Arsenal on Saturday, are bottom of the table with 20 points from 22 games and are in desperate need of points but Grant believes nothing will be decided in January.
"I think we need to look in the mirror and see what we are now," he added.
"I don't say anything against the people in the past but our target is very clear, not to be relegated.
"I believe very much in this club. But we were not Barcelona in the last year and the years before."
Grant also mentioned that 35 points, which ensured survival last season, will not guarantee safety this term and predicts many twists and turns in the coming months.
"We will have many changes in the table," he said. "Nothing will be decided this month.
"It is not just football, it's about spirit. There will be seven teams or more [involved in the relegation battle].
"Last season it was 35 points [to survive], this season I don't think 35 will be enough.
"I don't know how much more but it's very important to be strong against the teams around you."
The Hammers go into Saturday's match against the Gunners have only lost once in their last seven matches and Grant admits he's seen improvements in his side.
"We are developing," he added. "Three months ago we were playing good football but football is about results.
"We are improving also because the spirit is very good, which you saw the other night. I feel every game we can win."