Transportation Secretary Drew Lewis is trying to sell the rest of the administration and key members of Congress on a 5-cent-a-gallon gasoline tax increase to restore the highway trust fund to solvency and create an earmarked source of federal aid to transit systems.

The Department of Transportation proposal would raise the at-the-pump tax on gasoline from 4 cents per gallon to 9 cents, or raise the same amount of money by increasing other user fees such as excise taxes on tires, according to well-informed sources. That would add about $5 billion annually to the highway trust fund balance.

About $4 billion of the new revenue would be devoted to highway projects, including completing the interstate highway system and maintaining what has been built. The other $1 billion would be used to start a special account for mass transit projects, with emphasis on rebuilding subway systems and purchasing or rehabilitating buses.

There would no change in the administration's plans to eliminate federal aid to the annual operating budgets of mass transit systems. A three-year phaseout of that program is in its second year and most transit authorities are beginning to accept the cut as inevitable.

Costly highway projects such as the Westway in New York City, the precarious status of the highway trust fund and the rapidly growing need to rebuild roads and bridges have led DOT to conclude that the trust fund, no matter what its faults, must be saved. The fund's authorization must be renewed by Congress next year and the fund is not big enough to complete the interstate system alone, much less provide federal aid for other highway projects.

DOT officials, notably federal highway administrator Ray A. Barnhart Jr., have been making speeches about the need for a gasoline tax increase. While the White House position is not known, it has not moved to rein in Barnhart. Briefings by DOT aides on the proposal are set this week for White House, Treasury Department and Office of Management and Budget officials.

It is assumed that budget director David A. Stockman would bless a gasoline tax increase because it would fall in the category of "revenue enhancement." It has the political advantage of being saleable as a user fee--where the highway user is charged for the cost of building and maintaining the system--rather than as a straightforward tax hike.

The more difficult part of the argument is going to be in convincing highway interests that a transit fund, financed largely from the gasoline tax, is good for them. However, a growing number of state transportation directors have learned that good transit systems make it easier for cars to run on the highways.

Secondly, Lewis and his lieutenants are known to be alarmed at the extraordinarily high--and ever-growing--cost of urban freeways, which always seem to be full of cars during rush hour no matter how how wide the road. "We can never build enough lanes to handle it," one DOT official said.

By concentrating on rebuilding deteriorating subways in New York, Philadelphia, Boston and Chicago, and by continuing new rail programs such as Washington's, the federal government would get substantial lobbying support from contractor groups that are just as willing to pour concrete in tunnels as they are on roadbeds.

The highway trust fund, invented by the Eisenhower administration in 1956 to build the 42,500-mile interstate system, is now obligating about $11 billion annually, while revenues have leveled off at about $7.5 billion.

At the same time, the estimated cost of completing the interstate system alone runs between $38 billion and $53 billion. Further, the administration has proposed eliminating urban freeway and secondary highway programs from the eligibility list for federal aid, and there is growing pressure to save those categories.

In addition to the 4-cents-a-gallon federal gasoline tax, the fund derives revenue from special taxes on trucks and buses plus excise taxes on tires. There is a hot debate over whether trucks pay their fair share and Congress ordered a study on that the last time it renewed the trust fund, in 1978. That study, nearing completion, is certain to be an integral part of the debate.

A gas tax boost was proposed by the Carter administration as it was leaving town but the plan went nowhere. If DOT's package wins President Reagan's support, it will still have a tough time in Congress because it has to be sold to tax-writing committees as well as the friendlier DOT authorizing panels.