House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) plans to meet today with Budget and Appropriations committee leaders to find a way around the latest problem faced by Congress' six-year-old budget process.

At stake is the so-called "reconciliation" mechanism under which Congress must approve $6.4 billion in spending cuts -- plus raise $4.2 billion in additional revenue -- in order to keep the budget it adopted last week in nominal balance.

Only a day after both houses approved an initial budget resolution for 1981, the House Appropriations Committee last Friday brushed aside reconciliation instructions in approving $500 million to continue Saturday mail deliveries -- an expenditure that the House is expected to approve tomorrow.

Budget Committee Chairman Robert N. Giaimo (D-Conn.) quickly charged that the Appropriations Committee was jeopardizing the whole budget process.

But O'Neill yesterday defended Appropriations, asserting that the Budget Committee would be "transgressing on [its] rights" if it were to order specific spending cuts rather than set ceilings for various categories of spending.

It is a long-standing dispute involving whether the Budget Committee is to be "all-powerful" in spending matters, O'Neill said.

Reconciliation was included as a self-disciplinary tool for Congress in the 1974 Budget Control Act, but it has not been used until this year, when Congress had to employ it to shoehorn its spending plan into the relatively narrow confines of a deficit-less budget. While most experts believe the recession will drive the budget out of balance by the end of the summer, congressional rejection of reconciliation on its first key test could accomplish the same thing even sooner.