Sterling fell 1.5% against the US currency on Sunday to its lowest level since the flash crash in October.
Analysts said traders were reacting negatively to reports that the prime minister would signal plans on Tuesday to quit the EU single market.
The pound has now dropped about 20% against the dollar since the referendum when it fell to 31-year lows.
The pound also dropped to a two-month low against the euro on Sunday, falling to about â‚¬1.13.
Several of Sunday's newspapers claimed Mrs May would outline a "hard Brexit" approach, a term used to imply prioritising migration controls over single market access.
Downing Street has described this as "speculation".
However, reports of the UK leaving the single market are "like kryptonite" to traders who back the pound, said Kathleen Brooks, an analyst at City Index.
"The FX market has spoken, and, as of Sunday night, it is not confident that Theresa May can deliver the necessary clarity and confidence when she lays out her Brexit plans in a speech on Tuesday," Ms Brooks said.
The "Brexit theme as bad news for the pound is such an ingrained trend" now that every headline can  "generate another wave of selling", she added.
The pound has been volatile since the Brexit vote, partly due to uncertainty about the economic impact if the UK gives up its tariff-free access to the EU.
It reached its lowest level in the early hours of 7 October when a flash crash during Asian trading saw it drop below $1.18.
A report on Friday said the abnormally large swing was due to a range of factors, including the time of day.
Trading in the pound can be more volatile in Asian markets, when key sterling counter traders in London and other important Western markets are not operating.
The pound has fallen against the dollar to below $1.20 ahead of a key speech from Theresa May on Brexit this week.