President Reagan yesterday urged Republican members of Congress to show courage in defending his proposals in the coming budget fight, but promised he will also listen to their ideas for new cuts in the budget.

"There will always be room for improvement in any budget," Reagan said in a letter sent from his Camp David retreat to House and Senate Republicans. " . . . Where further savings can be found, or a better way of meeting agreed-upon goals can be worked out, I pledge my full cooperation to you, and I want to hear from you."

The genial tone of the letter to members of his party contrasted to the recent "put up or shut up" challenge to critics made last week during a trip to three midwestern states.

The president indicated flexibility in some budget areas, but his letter restated the administration's firm stand against cutting defense spending or tampering with the big tax cuts passed last year.

The president warned that on these two matters "there must be no such thing as retreat."

The president delivered a similar message earlier last week, but this time the message came in a public letter while members of Congress were back in their districts for the 10-day Washington's Birthday break.

"While you are home this week I urge you to listen to the real voices of the American people, not just the squeaky wheels," Reagan wrote.

Reagan's record budget proposal of $757.6 billion for fiscal 1983 and its companion $91.5 billion deficit were received poorly on Capitol Hill. Several Democrats have proposed alternate spending plans, and the administration has dismissed them.

The letter set down the White House concern at the prospect of unhappy Republican lawmakers wavering in their support for such a massive spending and deficit plan as they begin their 1982 reelection campaigns.

"I understand the nervousness that some members of Congress may feel in an election year," Reagan wrote. "The temptation is always strong to go for the easy option, the quick fix that may buy a little time but solves nothing in the long run . . . .

"Where we have honest differences, you can count on me to be a willing listener and a sincere partner. But this is no time for turning back."

Reagan mentioned his trip to Minnesota, Iowa and Indiana last week, and said, "Time and again I was inspired by the commitment and resolve of average citizens from all walks of life--farmers, factory workers, small businessmen and professionals pulling together through difficult times to make our country great again."

He said that "at every stop on my trip through the heartland, they urged me to hold firm to the course we have charted for recovery."

In another announcement from the White House yesterday, a two-day stopover in London was added to the president's European trip scheduled for June 4 through 10.

The president is to begin his first trip to Europe since taking office at the economic meeting of the leaders of the industrialized nations in Versailles June 4. After the conference, he is to fly to Rome on June 7 for a meeting with Italian President Sandro Pertini and Pope John Paul II.

From Rome he is to fly to London for a visit with Queen Elizabeth II before heading for the summit meeting of the North Atlantic Treaty Organization leaders in Brussels on June 9 and 10.