Deputy Tory leader Alistair Thompson has written to the council's chief executive David Williams saying he is 'increasingly disappointed' with the contract.
He says pavements and roads are not being resurfaced to an acceptable standard and claims councillors were not told by council officers what materials would be used before they agreed the contract.
Councillor Thompson wrote: 'I don't believe that the problem lies with Colas, who by and large have performed above and beyond what I expected. My concerns have been about the way councillors were told clearly by senior officers that the contract was on a like-for-like basis.
'Like for like means in my mind exactly that, Tarmac for Tarmac and paving slab for paving slabs and yet over the past few months we have seen a number of derogations from this original principle.'
The letter comes after The News reported complaints from residents about the use of microasphalt, or slurry seal. The 'temporary' pavement covering has been used by Colas to patch-up streets until they are fully resurfaced in up to 10 years' time.
Residents say the surface is not up to scratch and leaves pavements rough, with cracks for weeds to grow through.
Cllr Thompson called for a timeline of events leading up to the decision to use microasphalt. 'I must formally request that you set up a full independent inquiry into this matter', he wrote.
'In short, action needs to be taken to resolve this complete mess before the work of Colas and the creditability of contract are brought in to doubt.'
City council solicitor, Michael Lawther, said: 'We have received a letter from Councillor Thompson in the past few days outlining some concerns. We will now be taking steps to establish what the facts are regarding those concerns and it would not be appropriate for us to comment further until that has been done.'