(Recasts with Southwest signaling possible fleet trim)

DUBLIN/CHICAGO, Nov 19 (Reuters) - A day after Boeing Co received approval for its 737 MAX to fly again following a 20-month grounding, its two largest U.S. and European customers signaled caution on their order books as they monitor demand in the midst of the coronavirus pandemic.

U.S. based Southwest Airlines, Boeing's largest customer worldwide, said this week it would just take new MAX jets to replace jets it's retiring rather than grow its fleet, and on Thursday raised the prospect of scaling back its fleet due to the pandemic.

"If demand is going to be persistently depressed, we can retire and not need to take airplanes as replacements," Southwest Chief Executive Gary Kelly told journalists.

When the 737 MAX was grounded globally in March 2019 following two fatal crashes, airlines canceled flights because they lacked enough airplanes to meet strong travel demand.

Now many have parked jets or even gone out of business in the pandemic, creating challenges for Boeing as it tries to find homes for 737 MAX jets that are built but now lack buyers.

A senior executive of Boeing's largest European customer, low-cost carrier Ryanair, said on Thursday that it is not currently looking at additional orders for the jet, but may buy more if prices fall in the wake of COVID-19.

Executives at the airline have said repeatedly over the past year that the airline was discussing possible additional orders amid ongoing talks with Boeing on compensation for delays to its current order for 210 MAX jets.

"Over time one can see that there will be a way that aircraft are going to be cheaper ... and we will capitalize on that at some stage," Ryanair DAC Chief Executive Eddie Wilson said at the Skift Aviation Forum.

"But at the moment it's just the ... existing order that we have," he said.

Ryanair has said it expects to take delivery of its first 30 MAX jets by next summer. The United States lifted a 20-month-old flight ban on the MAX on Wednesday and European regulators are expected to follow suit.

Norwegian Air Shuttle, another major MAX customer, this week sought bankruptcy protection in Ireland. (Reporting by Conor Humphries and Tracy Rucinski; Editing by Edmund Blair and Nick Zieminski)