The Financial Conduct Authority (FCA) had set out proposals to improve competition in the cash savings market by introducing a "single easy access rate".
"Given the continuing impact of coronavirus and the low-interest rate environment, we have decided to stop this work," the FCA said in a statement.
The Bank of England has slashed its benchmark interest rate to 0.1%, with markets pricing in a move to negative interest rates. The gap between rates paid to new and longstanding customers has fallen, the FCA said.
Last year the FCA proposed a package of measures aimed at improving competition in the investment platforms market, including restricting the use of exit fees.
The watchdog had said it would give platforms time to make changes or it would consider further measures, including a ban on exit fees levied on customers that switch to another platform.
Two major platforms, interactive investor and Hargreaves Lansdown have said they would no longer charge them.
"We have therefore decided to stop the exit fees work but will be closely monitoring the situation, with the potential to consult on new rules if market changes lead to harm re-emerging in this area," it said.
Interactive investor said it was completely bewildered by the FCA decision and that exit fees should be banned given there were still some platforms relying on them.
"Scrutiny on exit fees needs to be extended to life companies, asset and wealth managers, life insurers and beyond," interactive investor CEO Richard Wilson said.
Hargreaves Lansdown said halting the consultation is a missed opportunity.
The FCA is also delaying until next year a public consultation on potential changes to the "duty of care" financial firms have to their customers.
(Reporting by Huw Jones; Editing by Kirsten Donovan and Emelia Sithole-Matarise)
By Huw Jones