More and more Britons are asking which side the Americans will line up on if the Falklands dispute comes to shooting. Privately, the British government expects the United States to behave like Britain's ally.

"If Haig fails," said a Cabinet minister with no diplomatic responsibilities, "the Americans have to come down clearly on the side of the British and what is right."

U.S. diplomats and journalists here are asked constantly--by British diplomats, politicians, journalists and others--whether Britain's closest ally would be on its side, diplomatically at least, if this country becomes involved in fighting with Argentina.

It is a hot topic in the British media, which has broadcast and printed at length much of the evidence from the other side of the Atlantic, from statements by Ambassador Jeane J. Kirkpatrick questioning whether technically Argentina had invaded the Falklands to the impassioned argument by NBC commentator John Chancellor that a U.S. government should never treat Britain and Argentina equally as "America's friends."

While admitting dismay about some statements by Kirkpatrick and President Reagan during the crisis, officials said Prime Minister Margaret Thatcher's government appreciated U.S. support of the U.N. Security Council resolution calling for Argentine withdrawal from the Falklands and U.S. suspension of arms sales to Argentina.

On Haig's role as negotiator, British Foreign Secretary Francis Pym said, "It seems to me quite reasonable that in that context he should try and maintain an even-handed position."

"But I have no doubt," Pym added, "that the spirit of the United States is in favor of democracy, which their country has always been, as has ours, as opposed to the Argentinian regime, which is not a democratic country, with a rather bad record on human rights. The U.S. spirit will all be there, and of course we are long-standing partners."