Johnson meets European Union leaders at the UN General Assembly in New York.
Expectations for progress were low, and the substantial volume of short positions built up on sterling may shield the currency from any sharp declines, analysts said.
"We are sceptical of much in the way of progress being made, and on top of that there is also the risk of PM Johnson having to cut his trip short to return to London if the Supreme Court decision goes against the government," MUFG analysts said in a note to clients.
Johnson suspended parliament for five weeks, until Oct. 14, a move critics saw as an attempt to stop MPs from preventing Britain leaving the EU on Oct. 31. Scotland's highest court of appeal ruled the suspension unlawful, but Johnson appealed.
The Supreme Court said it will issue its decision at 0930 GMT on Tuesday. A ruling against the government would increase political uncertainty and "weigh on sterling a little", said Jeremy Stretch, head of G10 forex strategy at CIBC.
The collapse of the British travel firm Thomas Cook could also put some pressure on the pound by highlighting the weakness of British retailing, Stretch said, adding that he sees "a drift lower for cable (pound/dollar) for now".
Sterling/dollar will likely trade around "$1.23 in the short term", Stretch said.
By 1445 GMT, the pound was down 0.3% at $1.2430, having dropped earlier to a six-day low of $1.2413, mostly because of dollar strength after a solid composite purchasing managers' survey.
The dollar index was up 0.2% at 98.68 after rising earlier to a week-and-a-half high of 98.832.
Against the euro, the pound was flat at 88.36 pence .
Hedge funds trimmed their short positions on the pound to $6.73 billion in the week to Sept. 17, according to the Commodity Futures Trading Commission, but the amount remained near the two-year high of $7.81 billion it reached last month.
Last Thursday, the pound reached a two-month high after European Commission President Jean-Claude Juncker said a Brexit deal was possible.
But sterling pulled back on Friday after Irish Foreign Minister Simon Coveney said London and the European Union were not yet close to a Brexit deal.
"In terms of Brexit news, the outlook for sterling is still binary," said Jane Foley, senior currency strategist at Rabobank.
"If optimism is extended, euro/sterling can be expected to push back below the 0.88 level this year. By contrast, if hopes that a deal could still be in place by the end of next month are snuffed out, euro/sterling could be re-setting its sight on the 0.90 level and potentially above," Foley said.