However, the CML warned that the precise level of the reductions would be a commercial decision for each individual lender.
Michael Coogan, director general of the CML said: "The problem banks have got is that they have limited funds and don't have enough money to give to all the customers who may want them.
"I think over the next few days and weeks we will see that the banks and building societies will move by anywhere between 0.5% and 1.5% - the individual decisions will be on the basis of assessing what they want for their savers as much as what they want for their borrowers," he added.
These cuts would be for existing customers on standard variable rate (SVR) deals, with deals for new borrowers likely to be re-priced.
Almost all tracker mortgages have been withdrawn for new borrowers as lenders consider at what rates to reintroduce them.
Conservative Party leader David Cameron said that banks that had taken part in the government's bail-out programme should be forced to pass on Thursday's interest rate cut.
Whereas it's wholly rational for any individual bank to take a much more cautious and conservative approach to lending, it's wholly irrational for all of them to do so at precisely the same time
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