May wrote to European Council President Donald Tusk on Friday asking to delay Britain's EU departure until June 30 to allow divided British lawmakers to agree a withdrawal deal.
Britain's exit from the EU, nearly three years in the making, is now in doubt because the British parliament cannot decide what exit terms it wants.
The impact of a long Brexit delay on sterling is unclear. The pound would gain if a delay led eurosceptic lawmakers to back the deal May negotiated with the EU or brought about a reversal of the 2016 Brexit referendum.
But on Friday sterling stuck close to $1.31 - the middle of its trading range for most of March - and traders said the path to any resolution remained murky.
"We favour a tortuous path to a deal leading to some eventual pound gains, with $1.2950/3000 remaining the lower end of the near term range for Cable," said analysts at ING.
Adding to the uncertainty, France said it considers premature talks of granting Britain another extension.
UBS Global Wealth Management said it foresees a long extension to Brexit, because it doubted May's attempt to build a cross-party consensus would succeed and expected the pound to remain volatile.
May is talking with opposition lawmakers to create a cross-party consensus to break the Brexit deadlock, but Labour leader Jeremy Corbyn said she had not moved far enough in a first round of the talks.
This week's developments have also pushed up expected gauges of volatility for the pound, with one-month implied gauges elevated vis-à-vis the euro.