Aides to Republican presidential candidate Rudy Giuliani refuse to speculate, but are evidently very excited about the prospect that the former New York City mayor has won the GOP fundraising race in the second quarter of this year.

Giuliani sources say it is possible but unlikely that they will have their final count on Monday, but they expect to match their first quarter fundraising total of $16 million, including general election money, and could possibly exceed it.

That seems to be enough to outraise Mitt Romney, who is not expected to match his first quarter take of $20 million. Romney is also pouring in some of his own money into this quarter's numbers, which will have to be deducted from his total for a clearer picture of donor enthusiasm.

Varying accounts seem to suggest that Giuliani's fundraising has improved while Romney's has declined. Romney aides point out that the former Massachusetts governor has focused all of his attention on Iowa and New Hampshire, where he leads in the polls, and has spent less time on fundraising.

Meanwhile, Giuliani, whose operation is much more organized than it was last quarter, has downplayed his standing in the nation's first caucus state, announcing that he will not compete in the Iowa straw poll that has traditionally been used to set the tenor for the January caucuses.

Arizona Sen. John McCain failed to match the $13.5 million he raised in the first quarter despite guarantees by senior advisers a month ago that he would reach $15 million. McCain's campaign announced Monday that it had collected $11.2 million, and was cutting staff and reducing salaries for other aides to compensate for the shortfall.

None of the top three Republican candidates are close to the numbers announced by Democratic Sens. Hillary Clinton and Barack Obama. Obama campaign officials announced Sunday that they are looking at a $31 million figure for primary election funds raised in the April-June quarter, outpacing the New York senator and former first lady by $10 million in primary spending cash.

FOX News' Carl Cameron contributed to this report.