LONDON/SINGAPORE (Reuters) - The U.S. dollar fell on Tuesday while the euro rallied as stocks rebounded in Asia and Europe on hopes that U.S. President Donald Trump will enter negotiations over his sweeping tariffs that have roiled markets for three days.

Rises in both the Japanese yen and Swiss franc showed consistent appetite for safe-havens, however, as investors remain concerned about the potential for a global recession.

Currencies have been highly volatile in recent days as investors tried to work out which offer safety while markets sell off, and which economies might be worst hit.

On Tuesday the euro was last up 0.3% at $1.0931, down from an earlier rise of more than 0.7%, after falling for the two previous days.

Currencies that often fare well when stock markets are rising also recovered, with the pound up 0.2% and the Australian dollar 1.1% higher after both dropped in the previous two sessions.

"Sentiment is rebounding, perhaps on the view that Trump may focus protectionism on China and speed up trade deals elsewhere," said Francesco Pesole, currency strategist at ING. "Markets may be erring on the optimistic side though."

Investors on Tuesday gleaned some positive signs from the Trump administration about tariff talks. Treasury Secretary Scott Bessent said on Monday he hoped negotiations would bring levies down.

Trump said Japan was sending a team to start negotiations, helping Japanese equities rally sharply overnight.

However, China dug in and criticised what is called "blackmail" from the United States over Trump's threat of additional 50% tariffs in response to China's initial retaliation. Meanwhile the European Union floated 25% counter-tariffs on U.S. goods.

The dollar was last down 0.4% against the Japanese yen, traditionally seen as a safe-haven at times of market stress, at 147.28 yen to the dollar. The U.S. currency touched a six-month low against the yen on Friday.

Meanwhile the U.S. dollar index, which measures the currency against six peers, was 0.3% lower at 103.11.

It has fallen around 0.7% since Trump announced the tariffs on April 2, as investors have weighed up the hit to the U.S. economy against the currency's typical role as a shield from market slumps.

"The current volatility is entirely the result of the policy choices of the Trump administration, meaning that, if reversed, the impact on financial markets will likely reverse as well," said Nathan Lim, chief investment officer at Lonsec Investment Solutions.

China's yuan fell to its weakest level since 2023 after the central bank slightly loosened its grip on the currency in what analysts said was an attempt to counteract the blow to exports from tariffs.

The move "has added to building speculation amongst market participants that China could allow a bigger devaluation of the renminbi to offset the negative impact (of) the worsening trade war," said Lee Hardman, senior currency analyst at MUFG, using an alternative name for the yuan.