Prime Minister Menachem Begin today canceled a longstanding appointment he had for Tuesday with President Chaim Herzog, effectively delaying submission of Begin's official letter of resignation to Herzog until after the Jewish New Year holidays this week.

The cancellation of the meeting coincided with the increased difficulty that Begin's designated successor, Foreign Minister Yitzhak Shamir, is encountering in his attempt to hold together the existing government coalition. Shamir met with leaders of some of the other parties that form the government coalition for five hours today, but no progress was reported.

It is widely assumed here that the longer the negotiations drag on, the greater the danger to Shamir's chances of forming a new government. The foreign minister's hope of reaching a quick agreement that would hold together the existing coalition under his leadership appears to have been dashed and it is not clear how much longer the negotiations will take.

Begin's scheduled meeting with Herzog was to have been a traditional courtesy call marking the end of the Jewish calendar year. Ami Gluska, an aide to Herzog, said it was decided it would be awkward to go through with a courtesy call while the country is awaiting the more dramatic moment when Begin officially submits his resignation.

The prime minister announced his resignation last week but agreed to delay implementing it to allow his political colleagues time to settle on a successor. Three days later Begin's political party, Herut, elected Shamir as its leader, giving the foreign minister the party's mandate to succeed Begin.

The Herut endorsement, however, is only the first hurdle Shamir must clear if he is to become Israel's seventh prime minister. He also must reach agreement with the smaller parties that are part of the coalition and whose votes are critical to maintaining a parliamentary majority.

Last week Shamir said he hoped to present a new government to Israel's parliament, the Knesset, this week and to ask for a vote of confidence that would make him prime minister.

That now appears extremely unlikely. The Jewish New Year festivities begin at sundown Wednesday and last through the weekend. Begin's official resignation is now not expected until Sunday at the earliest.

The Begin government consists of six political parties--Herut and the Liberal Party, which together form the Likud bloc, and four smaller parties. Together this coalition controls 64 votes in the 120-member Knesset.

Shamir met today with representatives of two of the smaller parties, TAMI, whose constituency is largely of Israelis from North Africa, and Agudat Israel, the guardians of religious orthodoxy in the country.

Both parties have been dissatisfied with their treatment in the Begin coalition, and before the prime minister announced his resignation TAMI was threatening to bolt from the government because of new economic austerity measures. TAMI leader Aharon Abuhatzeira was quoted today as saying the party would not join a coalition headed by Shamir unless the economic measures are rescinded.

The threat illustrated some of the difficulties facing Shamir. To win TAMI's support he may have to back down on some of the economic issues, but if he does so he may weaken his ability to get control of the troubled economy.

Last week the leaders of all the coalition parties signed an agreement in principle to remain in a Shamir-led government, but the details and additional concessions each will demand were left open for negotiation. The agreement will also not prevent the smaller parties from holding talks with the opposition Labor Alignment. Both TAMI and the National Religious Party reportedly have scheduled meetings with Labor officials Tuesday, although at the moment there does not appear to be a serious prospect that they would agree to join a Labor government.

There also appears to be a fight building over the role of former defense minister Ariel Sharon in the next Israeli government. There have been reports that, in return for Sharon's support within the Herut Party, Shamir has promised to put the former defense minister in charge of a key government committee that oversees Jewish settlements in the occupied West Bank and Gaza Strip. Sharon, a controversial politician here, was forced to give up his post as defense minister by the findings of an Israeli commission investigating the massacre of Palestinian refugees in a Beirut camp last year.