Dr Pete Sudbury, a Green Party councillor, told the meeting: “It is performative cruelty.
“That beautiful ‘don’t let them come legally and then punish them for coming any other route’ is a true stain on this country’s reputation and I am sorry.”
Conservative group leader Eddie Reeves told the BBC: “I agree that it would be better for asylum seekers to feel that they could get around locally and contribute positively to society while their claims are being processed.”
The Labour and Reform parties have been approached for comment.
Asylum seekers receive £8.86 a week if their accommodation provides their meals.
Oxfordshire County Council said the issue of free bus passes was “complex”, while the Department for Transport declined to comment.
A free travel scheme in Wales ended in March, while Scotland provides free travel to asylum seekers who have a qualifying disability or are aged under 22 or over 59.