The 27-year-old Bournemouth forward, who joined City on loan in January and scored six times in 18 games, has a contract with the Cherries until 2018.
"We're in ongoing discussions," Ashton told BBC Radio Bristol.
"We'd all love to see Lee Tomlin here. But what we've got to build here is a squad, and it's not about one player."
"We've been in constant contact with Lee throughout the summer," Ashton added.
"But there are so many different factors to bringing him here that would have to add up to make it happen, it's ongoing. It has to fit everything else that we're doing."