House Democrats expressed increased support yesterday for raising taxes on the wealthy to meet budget targets, but reached no agreement on the specifics of the tax bill they will begin writing as early as next week.

With Republicans likely to oppose tax increases almost unanimously, Democrats must come up with tax-increase legislation that can gain the votes of a large majority of their party.

President Reagan has vowed to veto any tax increase. Although some lawmakers believe there are certain kinds of tax increases Reagan might consider, a tax bill with a soak-the-rich flavor is not among them.

During a meeting of about 100 House Democrats yesterday, "several speakers talked about making sure whatever we do hits the rich and corporations," said Rep. Marty Russo (D-Ill.). "That was one thing you can say was a thread running through a lot of speeches."

Specifically, legislators called for postponing tax-rate reductions scheduled for next year, a move that is strongly favored by House Speaker Jim Wright (D-Tex.).

Ways and Means Committee Chairman Dan Rostenkowski (D-Ill.), who underwent surgery last week and was not at the meeting, remains opposed to altering last year's tax-revision law by freezing rates at this year's level.

Wright said there is "a very strong consensus" for taxes aimed at "those people who benefited most from the Reagan {tax} cuts and those whose reductions contributed the most to the deficit."

The speaker said an oil-import fee is unlikely because of opposition from New England lawmakers and a higher gasoline tax would encounter oil-state opposition. "A lot of people are saying, 'Don't give us a package of excise taxes that fall on consumers, give us a tax based on the ability to pay rather than an inability to escape,' " Wright said.

In what appeared to be a veiled reference to Rostenkowski's opposition to rate changes, Wright said Democrats are "wide open" to delaying rate reductions for high-income taxpayers, but added that the Ways and Means Committee opposes it. "Everything's supposed to be on the table," Wright said.

The congressional budget resolution requires that an additional $64 billion in revenue be raised during the next three years, $19.3 billion of it in fiscal 1988.

But many Democrats who voted for that resolution have said they do not feel bound to support a tax bill that raises the requisite revenues.

One option that garnered some support yesterday was a proposal by Rep. Byron L. Dorgan (D-N.D.) to catch more tax cheats by increasing resources for various functions of the Internal Revenue Service.

While tax-increase bills often fall back on compliance measures to raise revenue, the Dorgan plan was prepared by a large panel of tax specialists and is more comprehensive than most enforcement proposals.

It has the added political benefit of raising revenue without actually increas- ing taxes, an advantage that legislators suggested may induce the White House to consider it.

Dorgan said his package could increase tax revenue by $7 billion in 1988 and $42 billion over three years, although official congressional estimates are likely to forecast smaller increases.

Several Democrats said after the caucus that prospects for a tax increase this year remain uncertain. Rep. Steny H. Hoyer (D-Md.) pointed out that Reagan's avowed opposition to higher taxes means that "if the alternative is not presented well, we lose." And Dorgan said the Ways and Means panel probably will begin its drafting sessions next week "without anything locked into place. I think the committee will write a tax-increase bill in a process that will be pretty agonizing."