When Alexandria Mayor James P. Moran Jr. was in New York City last week, some City Council members talked about eliminating his job.

Nothing against Moran, they said; in fact quite the opposite.

The mayor's duties have proven so time- consuming that Moran has said he is going broke in the $12,500-a-year post because he can work only two or three hours a day at his job as a stockbroker.

Why not forget about electing a mayor and just choose one of the council members as chairman for a year, suggested council member Carlyle C. Ring. Arlington runs its County Board with a chairman elected for a one-year term.

Ring, a Republican, may have had more in mind than ensuring restful days for Moran, a Democrat.

In more than a century, no Republican has been mayor of Alexandria. But the GOP is only one seat away from controlling the seven-member council. If it can win one more seat the next election, it would have the votes to appoint a Republican chairman under the Ring proposal.

Alexandria's GOP chairwoman, Jane Ring, who is Carlyle Ring's wife, likes his idea of letting the council elect its own chief every year or so. The idea, she said, "certainly does have some merit."

Carlyle Ring's informal proposal, which Democratic and Republican members have agreed to discuss after the 1987 budget is adopted next month, would require a change in the city charter and that would require approval from the General Assembly. It could affect only future councils.

The likelihood of a GOP majority on the City Council will probably be better in 1988 than any time in the recent past. The party has scored steady gains in city elections, and last May three Republicans came within 1,100 votes of unseating Democrat Lionel Hope and defeating another Democrat, Redella (Del) Pepper.

Not unexpectedly, some Democrats do not care for the Ring proposal. "I think it's terrible," said Joann Miller, the city's Democratic Party chairwoman. "We're just out of one election and we're starting another."

Miller said she favored keeping the mayor's race separate from the other six council contests because of tradition and the desire to keep the mayor separately accountable. Further, she said, if the council elected its own chief, "It would really politicize the council."

Despite his financial difficulties, Moran is not for the idea, either. "I don't think it's really what citizens want," said Moran on his return from New York, where he had been on city business.

The mayor said he believed that Alexandrians wanted to choose the one person who would represent them for a certain period, instead of allowing the council to pass around the ribbon cutting, citizen complaint, staff and economic development duties.

In the 1940s the Alexandria council did choose the mayor from its ranks, and in 1948 the highest vote-getter on the council was automatically named mayor. Then in 1958, the mayor's race was separated from the rest of the council races.

"One of the criticisms has been there has been too heavy a burden on the mayor," said Carlyle Ring. Although under the city charter the mayor has no mandatory administrative duties and only one vote of seven on the council, Ring said there is a "perception that he has more authority than he does."

That perception was fed, no doubt, by former mayor Charles E. Beatley, who was retired and worked full time as mayor when he held the post.

Moran says he works at his mayoral duties 60 hours a week. As a stockbroker at A.G. Edwards & Son Inc., he works on a commission basis and says he has taken such a loss that he says he does not know if he can afford to run again or to put his two children through college.

Moran proposes that the council raise the mayor's and council members' salaries instead of shifting mayoral duties to a different person each year. Currently, council members and their aides are paid the same as Moran.

Ring rejects that proposal, arguing that any higher pay might encourage the "wrong kind of people."

Moreover, Moran said, Alexandria is a city that identifies with a mayor. Of the nation's 862 cities with a population over 30,000, about 75 percent elect a mayor, according to Lee Clay of the U.S. Conference of Mayors in Washington, D.C.

Council members Margaret B. Inman, a Republican, Patricia S. Ticer, a Democrat, and Pepper said they leaned toward a salary increase, although probably modest, and they said they wanted to discuss pay raises along with Ring's proposal after May.

With Montgomery County Council members considering increasing council members' annual salaries from $37,347 to $48,000 and Fairfax supervisors talking about a Chamber of Commerce recommendation that they raise supervisors' salaries from $21,590 to $35,000, the climate may be ripe for approving the politically sticky issues.

Still, there are many people around City Hall who are saying that this issue could well be decided on a 4-to-3 vote. For the moment, that likely gives the edge to Moran and his arguments for a better-paid mayor.