It comes as large numbers of visitors flock to the gallery to see a year-long exhibition of work by Damien Hirst.
A decision on who will host the Turner Prize is expected later this month.
Council chiefs today said it would be a real coup for the gallery and for the town.
Mike Bird, leader of Walsall Council, said: "It is quite a prestigious award, and if we could get it in Walsall that would be great.
"We have had a visit from people from the Tate and we now have to wait and see what the outcome is.
"The New Art Gallery has got a fantastic reputation and is attracting some big name artists."
Councillor Bird said he believed three or four other places were on the shortlist.
The gallery, which opened in January 2000, has recently reopened on Sundays – a move said to be crucial in its chances of hosting the annual art award.
Visitors have been able to go to the gallery on Sundays since the end of September and it is already attracting around 250 extra people every week.
Bosses also believe the year-long Damien Hirst exhibition will help put Walsall on the map.