On top of the tax increases and benefit cuts on which they have agreed for the short run, a majority of the president's Social Security Advisory Commission is expected to recommend this week that the normal retirement age be raised from 65 to 66 beginning early in the next century.

Senate Finance Commitee Chairman Robert J. Dole (R-Kan.), a member of the commission, said yesterday that he will back the age increase and he expects a majority of the commission to do so, perhaps as many as 10 of the 15 members.

A bipartisan commission majority agreed Saturday night on steps that would provide $169 billion over the next seven years to solve the short-term financing problems.

But that compromise, endorsed by President Reagan and House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.), would wipe out only about two-thirds of the system's long-range deficit. The bipartisan majority could not reach agreement on the remaining one-third, and it was agreed that the 15 members would make their own individual recommendations on this final issue.

Social Security's short-range problems derive mainly from the high inflation and unemployment of recent years. The longer-range problem grows out of demographics: the population is aging and by early in the next century there will be fewer workers per retiree.

Sources said yesterday that in the final commission report due later this week, the five Democrats appointed by O'Neill and Senate Minority Leader Robert C. Byrd (D-W.Va.) are expected to recommend imposing an added Social Security tax in the year 2010 of about about 0.46 percent each on employers and employes.

Most or all of the remaining 10 members appointed by Reagan and the House and Senate Republican leaders are expected to propose raising the normal retirement age--the age at which a person can retire and qualify for full Social Security benefits--from the present 65 to 66 in the year 2015. After that, the age would be gradually increased as average life expectancy rose.

However, congressional Democrats are expected to fight this proposal.

Despite its broad backing by top political figures, a "hard legislative fight" to pass the bipartisan compromise is expected because of the opposition of federal employe groups to inclusion of new federal workers in Social Security and of many others to partial taxation of benefits paid to those above certain income levels, the six-month delay proposed in the cost-of-living increase for this year and thereafter and acceleration of Social Security taxes.

Dole said he is eager to move fast on the proposal and will introduce it in legislative form next week, but does not expect to start markup until after the House Ways and Means Committee, which is scheduled to start hearings Feb. 1, has had a chance to proceed.

Yesterday the president spoke with O'Neill, House Ways and Means Chairman Dan Rostenkowski (D-Ill.) and Senate Majority Leader Howard H. Baker Jr. (R-Tenn.) in a conference call, and Dole later, to tell them he deeply appreciated their cooperation.

Dole said, "President Reagan told me today that he will actively support and work for the passage of this bipartisan agreement."

Baker said later he and O'Neill had discussed trying to speed up the legislative process as much as possible, and Baker added that while the bipartisan proposal could ignite "a firestorm of controversy," it would pass eventually.

Dole and House Rules Committee Chairman Claude Pepper (D-Fla.), a commission member, signaled yesterday that they have begun marshaling support for the agreement, Dole by "making calls to business groups to try to get endorsements" and Pepper, aides said, by getting in in touch with organizations of the elderly to get their support.

Sen. Russell B. Long (D-La.), senior Democrat on the Finance Committee, commenting for the first time on the bipartisan package, called it "a vast improvement over inaction" but added pointedly and without further explanation,"I am sure there will be ways of improving it."