The former England spinner has enjoyed plenty of success since taking over at Edgbaston, leading the team to the County Championship title with a victory over Worcestershire earlier this week.
"I think Ash would be the ideal candidate for England," Welch told The Telegraph. "Ash knows the England scene well, he has his own ways but he has instilled the discipline that he got from playing for England into our side.
"He's got a good eye for a cricketer and he's not hidden the fact that he would love to do it and I actually think that he would do a very good job.
"You have got to cut your teeth as a coach somewhere and it's difficult to learn your trade at a big club like Warwickshire. But Ash has done an excellent job for us. He works tirelessly, he loves talking cricket."
Warwickshire will be looking to win their second piece of silverware against Hampshire in the CB40 final on September 15.