The Republican party said it is open to a conditional short-term increase in the debt ceiling that would allow the US to avoid default.
London's FTSE-100 Index has been at a three-month low on the back of the recent uncertainty but climbed along with other European markets yesterday to stand 92.6 points or 1.5% higher at 6430.5.
The debt impasse has closed US government services ranging from museums to parks, but the Republicans' offer will raise hopes that America can lift its debt ceiling before a looming deadline of October 17.
The Dax in Frankfurt and the Cac 40 in Paris both climbed by about 2%.
On the currency markets the pound made modest gains against the dollar to stand at 1.60, while sterling was broadly flat against the euro at 1.18.
On the London market there were gains across the board, with high-profile risers including British Airways owner International Airlines Group following an improvement of 11.9p to 338.6p while insurer Prudential was up 39p to 1177p.
The Bank of England's decision to keep interest rates at a record low and maintain its quantitative easing programme at £375 billion came as little surprise to markets.
But households were still feeling pain after SSE announced that gas and electricity tariffs will rise by an average 8.2% from November 15.
SSE shares rose by 1.5% at one point before closing down 5p at 1449p. British Gas owner Centrica was 1.8p higher at 362.6p on expectations it will follow suit in the coming days.
Charles Church housebuilder Persimmon was the biggest FTSE-100 riser amid forecasts that it will reap the benefit of the Government's extended Help to Buy housing scheme. Shares rose another 5.7% or 64p to 1195p, having climbed by 5% on Wednesday.
It was joined high up the risers board by Costa and Premier Inn company Whitbread, which rose 122p to 3114p ahead of interim results later this month and fresh speculation of a break-up of the group.
In a strong session for shares in the leisure sector, Frankie & Benny's owner Restaurant Group rose 27p to 545p.
And Ladbrokes was 2.9% higher, up 5.2p to 185p, after fresh takeover talk was ignited by the mystery purchase of a near 3% stake in the struggling firm.
The biggest risers on the FTSE-100 were Persimmon up 64p to 1195p, GKN 15p ahead to 358.2p, Legal & General 8p higher to 194.5p and CRH up 60p to 1503p.
The biggest fallers on the FTSE-100 were Vedanta Resources down 5p to 1015, SEE 5p lower to 1449p, GlaxoSmithKline 4.5p weaker at 1545.5p and Shire off 6p at 2381p.