West Midlands Metro chiefs halted the service after the trams' manufacturer identified cracks in the underframe structures.
It meant a total of 21 trams were have been out of action since on Friday morning, and will remain so until further notice as inspections get under way.
The move has come days before the introduction of the city's Clean Air Zone, in which drivers are being hit with fees if they drive polluting vehicles.
Now the leader of the Conservative Group on Birmingham City Council has called for the scheme to be delayed until trams are running again.
Councillor Robert Alden, leader of the Birmingham Conservative Group, said: "This latest news just compounds the difficulties local businesses and workers are going to face from Labour’s travel tax.
"If people cannot travel into the city centre on one of the key pieces of public transport and are going to face a charge of £8 a day to use their car then it is simply going to punish those who have to travel for work and put off those people who would otherwise spend their money with retail and hospitality businesses struggling to rebound from the Covid pandemic.
"We have previously called for the charge to be delayed for a much longer period to allow businesses to recover, a request that has also been made by organisations such as the Southside BID. Labour are not willing to consider this, but surely now the least they can do is wait until the Metro is fully operational so that this at least is a viable option."
West Midlands Metro will make a decision on Monday, when the inspections are expected to finish, whether to resume its services, whilst passengers have been signposted to bus services, West Midlands Railway and Avanti railway services.
Councillor Ewan Mackey, deputy leader of the Conservative Group, said delaying the scheme – which charges car drivers £8 daily or £50 for HGVs and coaches – was the "right thing to do".
⚠️ #WMMUpdate [10:30] All trams services are suspended between Wolverhampton St George's and Library due to a fault requiring inspection of the tram fleet. Tickets being accepted on @nxwestmidlands 74 and 79 & @AvantiWestCoast & @WestMidRailway trains between Wol'ton and B'ham. pic.twitter.com/NsMdxJQ0ig— West Midlands Metro (@WMmetro) June 12, 2021
Metro tickets and passes will be valid on some alternative modes of transport including some bus services, West Midlands Railway and Avanti train services.