President Chaim Herzog today began the process that will lead to the formation of the next Israeli government.

Herzog consulted with leaders of the main opposition Labor Alignment and the ruling Likud bloc about their prospects of achieving a majority in Israel's parliament, the Knesset. Herzog will continue the process Monday and Tuesday with meetings with leaders of the smaller parties in the Knesset before asking one party leader to form a government to replace that of Prime Minister Menachem Begin, who resigned Thursday.

Both the Labor Alignment and the Likud bloc are seeking the right to form the next government, which must have an absolute majority in the 120-member Knesset. Under Israeli law, Herzog is free to ask any party leader to try to form a new government, but by tradition he is expected to turn to the leader he believes has the best chance of achieving a parliamentary majority.

That is expected to be Foreign Minister Yitzhak Shamir, who has been elected the new leader of Begin's Herut Party, the dominant party in the Likud bloc. Shamir last week also announced he had reached agreement with leaders of the other parties that make up the current government coalition to preserve the coalition's 64-seat majority under his leadership.

If the agreement holds up through last-minute political maneuvering, Herzog would appear to have little choice but to ask Shamir to form the next government. Herzog is expected to announce his decision Wednesday.

Neither Shamir nor Labor Alignment leader Shimon Peres was part of the party delegations that visited Herzog at his official residence today as the consultation process began.

Meanwhile, Begin remained secluded in his nearby official residence and failed for the second consecutive Sunday to attend the regular weekly Cabinet meeting. Deputy Prime Minister David Levy presided at the meeting, during which the Cabinet discussed the situation in Lebanon but disclosed no details of its deliberations.

Begin also failed to attend Yom Kippur synagogue services yesterday because of his apparently deteriorating health.