On the very day prime Minister Menachem Begin opened what are described here as "most difficult" talks with President Carter in Washington, the Israeli leader suffered a serious political setback at home.

It became clear here yesterday taht Begin's personal choice for president of Israel will not be elected when the 120-member parliament gets down to choosing the successor to President Ephraim Katzir in about two weeks.

The man most likely to be selected for the largely ceremonial position is Dr. Elimelech Rimalt, 70, whose candidacy was advanced by the Liberal party.

The Liberals are a part of Begin's Likud bloc in the ruling coalition.

Observers here described the independent action by the Libersals, led by Finance Minister Simacha Ehrilich, as a well planned blow to Begin's leadership that may eventually lead to the breakup of the Likud and the fall of Begin's government.

The very fact that the presidential candidate personally backed by Begin will not make it is seen here as countries other than Europe and America.

Having failed to come up with a suitable candidate who is politically close to Likud, Begin proposed a totally unknown candidate, Yitzhak Chavet, a nuclear physicist, who was born in Cairo and came to Israel in 1951.

While Chavet gave the impression of being a modest and sincere person, he failed to win the support of various groups within Begin's coalition and was received with even less enthusiasm among the opposition members of the Knesset.

"He has much to be modest about," one Labor leader quipped.