As tax revision flounders in the Republican-controlled Senate Finance Committee, congressional Democrats show signs of trying to exploit the issue politically.

The Finance Committee is to start closed-door meetings today that may revive the moribund tax measure, which Chairman Bob Packwood (R-Ore.) put on hold Friday to keep senators from passing amendments bleeding the bill of tax revenue.

Packwood has said he wants to use the private sessions, which are to exclude staff aides and Treasury Department officials as well as lobbyists and the public, to set broad policy goals for tax revision. Packwood said he hopes that those goals will keep the panel from deserting the path of "reform" as it tries to approve a tax bill in the next few weeks. The committee took no action on taxes yesterday.

Meanwhile, Democrats are looking to a potential election-year opportunity: a claim that the Democratic-controlled House passed President Reagan's top domestic initiative, a bill that would make wealthy corporations pay more taxes and poor people less, only to see it overturned in the GOP-dominated Senate.

"I think there's still a chance the Republicans will shoot themselves in the foot on this tax-reform issue," Sen. Bill Bradley (D-N.J.) told reporters at a breakfast yesterday. "I think it could be the dumbest thing they could ever do, but they may do it . . . . To the extent they do, they have given us an economic growth issue for the next decade."

At this early stage in the campaign season, Democrats are doing little more than talking about an issue they think could become politically potent.

In their party's response to Reagan's weekly radio address last Saturday, for example, Rep. Robert T. Matsui (D-Calif.) said:

"I hate to tell you, Mr. President, but if tax reform is the second American revolution, your troops in the Republican Senate have deserted you. They've deserted you, and they've deserted the poor and middle-class Americans who would benefit from tax reform."

Similarly, House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) said Tuesday that he would not accept Packwood's proposal to raise excise taxes on gasoline, alcohol, tobacco and other goods and services because the resulting higher prices would disproportionately harm poor Americans.

However, many Finance Committee members voting to preserve tax breaks are Democrats, and the tax-revision effort remains bipartisan.

In the closed meetings, Packwood is expected to try to form a "governing coalition" of senators from both parties that would support the broad goals of fewer deductions and lower rates.

Tax revision "is a problem whichever way we go, for some people," said Sen. George J. Mitchell (D-Maine), a Finance Committee member and chairman of the Democratic Senatorial Campaign Committee. "Some Democrats want it and some don't . . . . we ought to be concentrating on getting a bill out."

Voters have shown little interest in tax revision as a political issue. Even in Oregon, where Packwood is running for reelection, little attention has focused on whether he can extract a bill from his committee.

GOP proponents of tax reform have expressed concern that other Democrats will reiterate Matsui's charges frequently and loudly if the committee fails to produce a bill.

"President Reagan understood the attraction of this issue to middle-income voters . . . but precious few other Republicans understand it," Rep. Richard A. Gephardt (D-Mo.) said.