President Reagan's televised Monday night speech in behalf of his tax cut bill lit up the switchboards on Capitol Hill yesterday in the kind of public outpouring the White House claimed could well tip the scales for the president when the House votes today.

Early in the day both the White House and House Democrats said on the strength of headcounts that a rival Democratic tax bill was still ahead, though only a handful of votes. But by nightfall some Democrats were saying privately they were unsure their narrow majority would hold, and one Democratic leader said the Republicans could count on 38 Democratic votes -- enough to put them over the top.

The final jockeying came as the Senate virtually completed action on its version of the tax cut bill, on which it has been working for 11 days. (Details, Page A3)

Reagan kept up the tax pressure on the House yesterday, meeting with 32 more members and making a telephone call to a radio station in the district of one, Democrat Ralph Hall of Texas.

Members of Congress on all sides reported their offices at home and here flooded with calls, running as high as 6 to 1 in the president's favor. Western Union reported that telegrams to Washington were running at least 10 times normal volume by late afternoon, not including the slower Mailgrams.

By last evening the White House said it had received more than 4,000 phone calls and nearly 1,200 telegrams and Mailgrams on the tax issue, the calls 6 to 1 in favor of the president's position, the telegrams 10 to 1. The biggest single previous response in a 24-hour period during this administration was slightly more than 1,000 phone calls after the president's televised February speech in behalf of his budget cuts.

After an "inundation" of calls from constituents yesterday, Rep. Carroll Hubbard Jr. of Kentucky, a conservative Democrat who earlier voted against Reagan on the budget and who was counted as a swing vote on taxes, announced he will support the president. With 480 out of his 500 callers on Reagan's side, Hubbard said, "It is obvious that the president's tax cut plan has overwhelming support in western Kentucky."

Some Democrats grumbled that the blitz, orchestrated by the Republicans, came mainly from the upperincome individuals and businessmen who they said would be the main beneficiaries of the Reagan bill. House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.), still predicting victory for the Democrats, shrugged off questions about the outpouring.

"Three people hung up on me when I asked them if they made more than $50,000 a year," he cracked about what he said was only a trickle of calls and cables to his own office.

But Rep. Barbara Mikulski (D-Md.), a strong Democrat and opponent of the president's plan, had received over 360 calls by 1 p.m. "overwhelmingly" favoring the president's plan, according to aide Nick Glyphis. The calls came from middle-class suburban areas which were "not necessarily Republican but had voted Republican . . . My guess is it's spontaneous," he said.

In Democrat Morris Udall's office, calls from his Arizona constituents were running 6 to 1 in favor of the president, according to aide Pat Krueger, who was answering phones. "I'd say they are definitely from real people."

Rep. Beverly Byron (D-Md.), one of 15 Democrats wooed by the administration at Camp David last weekend, remained uncommitted despite phone calls running at five times the normal number, most in favor of Reagan. "Many are just John Doe constituents, but thee are also some corporations represented," said aide Jack Holliday. Byron "of course takes into consideration what her constituents express to her," he said. "But she will vote her concience."

Ellen Buchanan, an aide to Rep. Ken Holland (D-S.C.), who has opposed the president's tax plan after supporting his budget earlier, reported the successive waves of calls had forced the staff to "stop everything else just to answer the phone." But, she added, their calls were evenly divided. "People from more affluent areas support the president."

The eleventh-hour White House lobbying assault produced at least one mistake of a sort unusual in the Reagan operation, and more reminiscent of the Carter administration. Veteran legislative liaison Max Friedersdorf signed a letter which was sent to Arkansas Democrat Rep. Beryl Anthony Jr. The letter heartily solicited the support of one Rep. Anthony Beryl Jr.

The last-minute White House effort might have been damaged to some extent by the unexpected illness of Friedersdorf, who is hospitalized with an asthma attack.