The Senate Rules Committee gave narrow approval yesterday to an election-day registration plan that faces further obstacles and the threat of a potential filibuster on the Senate floor.

When President Carter proposed the legislation to Congress on March 22, it seemed so popular that even Republican congressional leaders were finding good things to say about the administration proposal.

Yesterday, the bill emerged from the Senate Rules Committee on a shaky 5-to-4 vote. Although no vote counts have been taken, administration sources are saying that they expect determined GOP opposition, several amendments and a probable filibuster on the floor.

The transformation of the bill's status from certain to insecure largely has been the work of the administration itself. Privately, Democratic senators are highly critical of Justice Department witnesses, who gave Republican opponents of the bill an opening by withholding a Justice Department memo that siad the measure would make it difficult to prosecute vote fraud.

The memo, when it was finally released, did not prove influential with either the Senate Rules Committee or the House Administration Committee. It is more a position paper than a reasoned analysis of the bill itself which in any event has been heavily amended since the memo was written.

But the memo has stiffened the resistance of elections officials in several states who are themselves worried about fraud possibilities.

Shortly before the Rules Committee approved the bill yesterday, Chairman Howard W. Cannon (D-Nev.) predicted that these officials would renew their efforts to defeat or postpone the legislation. A similar prediction on the House side came from Rep. Bill Frenzel (R-Minn.), the GOP expert on voter registration issues.

Despite the damage inflicted by administration witnesses, the measure still is favoured to win congressional approval because the majority Democrats regard the legislation as favorable to the Democrats and harmful to the Republicans.

"Regrettably, I think there's much more of a party lineup on this bill than anything else," said Frenzel yesterday.

This view, also is held by Rep. Frank Thompson Jr. (D-N.J.), chairman of the House Administration Committee. Thompson predicted yesterday that the bill would pass largely on party lines but said there would be "a very close vote" on a proposed amendment that would make the entire bill voluntary until the 1980 elections.

The administration strategy is for Thompson to bring the bill up for debate May 24, with a vote likely the following day. Frenzel said he will offer an amendment to make the bill completely voluntary.

Administration strategists believe that an overwhelming vote in favor of the House bill will help passage in the Senate, where a filibuster that would require 60 senators to stop is anticipated. The man who will be heavily involved on the Democratic side if this occurs is Vice President Mondale, whose understanding of the Senate twice rescued the bill when it ran into troubled in the Rules Committee.

Mondale's first service was in quickly securing release of the Justice Department memo withheld by Deputy Attorney General Peter Flaherty. His second was in persuading Sen. Claiborne Pell (D-R.I.) to support the measure and give it the margin it needed to win committee approval.

The Senate bill has some differences with the House version, which is considered the technically superior of the two. A principal difference is that the Senate would require indetification of election-day registrants by driver's license or similar method, while the House bill leaves this to discretion of the state.