Japan said on Monday it could end its state of emergency in many regions this week and New Zealand said it could ease restrictions on Thursday. The UK has also set out plans to ease the lockdown while in France shops re-opened on Monday.
South Korea warned of a second wave of the virus as infections rebounded to a one-month high and new infections have also accelerated in Germany.
"More of a risk-off tone has taken over at the start of this week," said Lee Hardman, currency strategists at MUFG, who said that there were some concerns that the re-opening of economies in places such as Germany may have led to a pick-up in infection rate.
Against a basket of comparable currencies, the dollar was last up 0.3% since New York's close, at 100.090.
The safe-haven Japanese yen hit a 10-day low versus the dollar, down around 0.6%, after a US buyer bought a large amount of dollar-yen, forcing the pair above 107.
Also weighing on global risk sentiment is the prospect of worsening tensions between the US and China.
A conciliatory phone call between US and China trade negotiators on Friday staved off fears of an imminent new round of US tarrifs. But US President Donald Trump said he was "very torn" over whether or not to end the preliminary phase one trade deal between the two countries.
On Monday, China warned that it will take countermeasures in response to a US decision to tighten visa terms for Chinese journalists. This news did not move the market, analysts said.
The euro fell against the dollar, last down around 0.3% at $1.08165.
"Developments in the euro area keep Eurozone equities and banks underperforming vs. the rest of the world, which does not help the euro," Morgan Stanley analysts wrote in a note to clients.
The riskier Australian dollar was down 0.7% versus the US dollar, while the New Zealand dollar was down 0.9% having fallen from around 0400 GMT.
The Swedish crown fell around 0.5% against the dollar, weakening to as much as 9.8150, also down around 0.2% against the euro, at 10.5985.
Minutes from the Riksbank's latest meeting, published on Monday, showed that Swedish rate-setters were united on seeing balance sheet measures as currently the best way of conducting policy amid the outbreak of the novel coronavirus.
"EUR/SEK is approaching attractive levels to buy, the pair is oversold and no longer reflects the likely central bank easing coming up. We suggest buying EUR/SEK on dips down to 10.55 (the long term support) and target 11.20," wrote Morgan Stanley analysts.
"All in, the factors which drove the EUR/SEK strength in the past eight years seem likely to continue to hold," they added.