The Republican quest for the White House in 1980 may well turn into a "Tale of Two Texans." Both men -- John Connally and George Bush -- are still long shots in the greatest political game on earth. But it's a good bet that one or the other will wind up on the ticket, as the running mate if not the presidential candidate.

It would be hard to find two men more opposite -- the brash, loquacious Connally, filled with Texas flamboyance; and the intellectual, laconic Bush, ruled by Ivy League reserve. They have widely contrasting styles, with Connally running a high-powered, wheeler-dealer, bigger-than-life campaign and Bush conducting a studied, low-key, grass-roots strategy.

Both Connally and Bush are "transplants." But Connally's political move from the Democratic to the Republican Party is likely to be more damaging to him in a national campaign than Bush's geographical move from New England to the Southwest.

Although both men are conservatives on major issues, they'll have their troubles with the hard-core right wing, which can be the spoiler at any GOP convention. Bush is considered a pin-stripe moderate by the right wing. And Connally, despite his recent alliance with the right-wing fund-raising whiz Richard Viguerie, is regarded by Republican hard-liners as nothing more than a "Lydon Johnson clone."

Their personalities are light years apart. The vain and vibrant Connally makes a virtue of his can-do image, presenting himself with supreme self-confidence as a leader who could master any situation instead of letting events put him on the defensive.

In contract, Bush is as determinedly reflective as Connally is extroverted. A lackluster public speaker compared with the ebullient Connally, Bush made no great splash in either his congressional career or his service as director of the Central Intelligence Agency and chief of the Peking diplomatic mission. Yet he is a quietly efficient administrator who can find his way through the dark tunnels of bureaucracy. He has an ability to put things in perspective and view them without emotion.

Connally has more ability to attract the national spotlight. He appears to be especially effective with Big Business audiences, to whom he offers such slogans as "We don't need more taxes . . . we need more energy." Connally is a one-man, non-stop, self-generating publicity machine.

Bush is so quiet that he is almost invisible; he scarcely registers in the political polls. He is trying to overcome this disadvantage by cultivating GOP professionals and grass-roots voters. He is zeroing in on New Hampshire, where he hopes to make a showing that will attract national attention. A knowledgeable political analyst recently in New Hampshire told our associate Jack Mitchell that Bush "has one of the best professional staffs of any of the GOP candidates." His chief advisers, David Keene and Jim Baker, are well thought of in Republican inner circles and are regarded as especially effective at nuts-and-bolts local organizing.

Connally operates more effectively at the national level than at the community level. While Big John rubs elbows with corporate fatcats at elaborate fund-raising functions, Bush goes on early morning jogs with local newspaper editors and precinct bosses in key primary states. Whether the money Connally raises or the contacts Bush makes will yield more delegates at the convention remains to be seen.

The attitude of the two men toward financial disclosure also underlines the difference in their approaches. Connally, the wealthiest of all GOP candidates, made more than $1 million in the last 18 months but refuses to disclose further details on his private financial holdings until nomination time. Bush, on the other hand, has opened his tax returns for scrutiny and disclosed his sources of income.

The two Texans' ties to former president Richard Nixon may also become a factor in the battle for delegates. Connally was admired by the deposed Nixon, who wanted to make him vice president after Spiro Agnew's fall. Nixon finally settled for Gerald Ford only because of doubts that Connally could survive Senate confirmation hearings.

Several Nixon loyalists have joined the Connally effort. But support from this quarter may turn out to be a mixed blessing; the Nixon taint may become an issue in the primaries. Although Connally boldly accentuates the positive aspect of his trial in the milk-fund scandal -- proclaiming himself the only candidate declared innocent by a jury -- he suffers from lingering suspicions. The jury list is still sealed, but we gained access to it and polled the men and women who acquitted Connally. Of the half-dozen who would talk to us, most merely gave him the benefit of the doubts they still harbor.

Bush was never a favorite at the Nixon White House, though he once served as GOP national chairman. Associates recall being startled by Bush's uncharacteristic bursts of anger at the time, shouting "No, I won't do it!" on more than one occasion when outrageous demands were made by Nixon's aides.

There are still many "ifs" in the Republican nomination fight, of course, with nearly a year to go until the convention. But if Ronald Reagan falters, Howard Baker vacillates and the trailing entrants fail to move, the eyes of the GOP may be on Texas.