* Dollar holds steep overnight gains
* Impeachment risk recedes, Trump stokes trade-deal optimism
* Sterling nurses losses as Brexit outlook clouds
* Graphic: World FX rates in 2019 http://tmsnrt.rs/2egbfVh
SINGAPORE, Sept 26 (Reuters) - The dollar found broad support on Thursday as investors welcomed U.S. President Donald Trump's hints of progress toward a trade deal with China and discounted the prospect of an impeachment probe making much headway in the short term.
The dollar index posted its sharpest daily gain in three months overnight and held steady in Asian trade.
The greenback also held gains on the euro and pound, steadying near two-week highs. A euro bought $1.0951 at 0105 GMT. The dollar hovered near a three-week high against the Australian dollar, which bought $0.6749.
"It's a pretty decent move really," said Nick Twidale co-founder of Sydney-based trade finance provider Xchainge, who said it was driven by optimism for a U.S.-China trade deal and markets' shifting the prospect of impeachment to the long-term.
"Markets shrug these things off very, very quickly and we move on to the next stuff. As it moves on, it's going to be interesting, but we have to go back to the fundamentals and fundamentals are going to push the dollar higher over time."
Sterling nursed losses at $1.2359, after dropping more than 1% overnight as investors priced in many more months of Brexit and election risk.
The greenback had been sold when U.S. House of Representatives Speaker Nancy Pelosi said the Democratic-led chamber was launching an impeachment inquiry over Trump's handling of a call to Ukranian President Volodymyr Zelenskiy.
A summary of the call subsequently released by the administration showed Trump pressed Zelenskiy to investigate former Vice President Joe Biden - Trump's leading Democratic rival - but did not show Trump threatening to withhold aid.
The House inquiry could trigger a trial in the Senate on whether to remove Trump from office, though that is seen as unlikely to succeed because Republicans control the Senate.
Trump also on Wednesday stoked hopes for a trade deal by telling reporters in New York that the U.S. and China were having "good conversations" and that an agreement "could happen sooner than you think".
"This was enough to turn the risk tide," said Michael McCarthy, chief market strategist at brokerage CMC Markets in Sydney of the president's comments.
It set trade-exposed risk currencies such as the New Zealand dollar and the Aussie drifting higher, though the strong dollar capped gains to below 0.1%.
Signs of a rebound in the U.S. housing market on Wednesday, when data showed August sales of new single-family homes rose more than expected, further buttressed the dollar. Few market-moving data releases are scheduled on Thursday.
Against a basket of currencies the dollar held around 98.959, just below a two-week peak. The safe-haven yen steadied at 107.71 per dollar.
The Chinese yuan was flat in offshore trade at 7.1278 after falling 0.3% overnight. ( Editing by Jacqueline Wong)