The White House has completed work on a tentative plan for distributing the $2.4 billion a year in windfall profits tax monies that President Carter has said he will earmark for grants to the poor.

The two-part proposal, now being circulated in Congress, comprises $2 billion a year in extra payments through the existing welfare payments through the existing welfare system and $400 million in cash grants to those hard-hit by rising hearing-oil prices.

The administration also is considering the possibility of proposing additional grants to help middle-income persons finance their higher fuel bills, but officials said that idea still is on the table and may well be scrapped.

The heating-bill aid for low-income persons would be distributed by the states, based on a special formula that would concentrate the funds in states with the coldest weather and the largest number of oil furnaces.

The basic outlines of the proposals have been known for several days, but officials provided new details yesterday on the mechanics of the plans. The White House intends to submit the measures formally when Congress returns.

The extra payments to recipients enrolled in existing welfare programs would be held to $1.2 billion in fiscal 1980, and expanded to a full $1.6 billion in fiscal 1981, which begins a year from this coming October.

Officials said the expanded program also may include a procedure to extend the added payments to poor persons who are not formally on the welfare rolls. These individuals would file applications with federal authorities.

Planners said the administration decided to channel the primary grants program through the existing welfare system mainly because it is already in place and can be adapted quickly to distribute the extra money.

The grants, which would come in the form of extra checks twice during each year, would provide an average of $100 for a single person or $200 for a family of two or more, to help with fuel and energy costs.

The smaller $400 million-a-year plan, to be administered by the states, would provide widely varying grants up to a maximum of $400 per family each year, based on the weather and heating-oil usage in the area.

For example, officials said a family of two or more living in Massachusetts would receive $248 under the plan, while the same size group living in Florida would get $120. Single persons would receive half those amounts.

The grants would be limited to persons with incomes below a complex "low-income" benchmark based on the family's Supplemental Security Income grant plus 125 percent of the present "poverty" level - a total of about $9,000 for a family of four.

The government has provided similar grants in the past two years, up to a maximum of $250 a family. The new version of the program would be set up to continue for four years.

The extra payments to be distributed through the existing welfare system would go primarily to two categories of recipients: Elderly persons who receive SSI checks, and families in the Aid to Families with Dependent Children program.

It was not clear immediately how Congress would react to the administration's proposals. Most key lawmakers were back in their home districts campainging, and were not readily available for comment.

The White House had promised to send the measure to Congress before the start of the August recess, but had second thoughts after Carter's energy tax-credit proposals took a beating from departing legislators.

Some congressional sources priased the administration for not proposing tax breaks as a way to distribute the monies - a move they feared would only open the door to a possible general income-tax cut bill later this year.

However, Capitol Hill sources predicted conflicts would arise over the plan to allocate the home heating-oil grants to states based on the temperature-and-oil-usage formula.

Although they conceded the plan made sense from a practical viewpoint, some of these sources raised the possibility that Southern and Western legislators might resent the concentration of the funds in colder areas.

Some consumer groups also have criticized the plan for not providing enough to cover the increases now expected in home heating-oil prices. Critics say in some Northeastern states the extra tab could top the average grant to the poor.