Sources tell CBS News Consultant Gloria Borger that the Gore campaign has "made its point of view very clear" to the White House.
Gore allies are worried that, when the president got into a back-and-forth with former President George Bush, it had no political upside for Democrats. The feeling was any such attack would only remind voters of the link between Gore and Clinton.
So they've asked the president to keep quiet.
On Tuesday, the rumble blew up when the Texas Gov. George W. Bush said he was "riled up" by Mr. Clinton's increasingly personal attacks.
Bush's father, the former president, told a breakfast for farm groups that his son is disciplined enough to "stick to the message" despite goading from the opposition.
"He can dish it out if he has to, but he's going to stay on his message," the elder Bush said of his son. "You don't believe me? Ask Ann Richards."
Richards is the Democrat whom the younger Bush ousted from the Texas governor's office, and who is remembered for deriding the elder Bush at the Democratic convention in 1988 for being born with a silver foot in his mouth.
"Well, she may have been right about that, but anyway, they cheered like hell and then George came along two years later and showed her what she could do with that silver foot," the former president said. "He did it by staying on the message."
On Tuesday, George W. said it was "amazing" that Clinton "would spend time trying to be a political pundit. He's so desperate to have his legacy intact by getting Al Gore elected, he'll say anything, just like Gore will."
Clinton has belittled the Texas governor's campaign, saying one reason he was in the running was because his "daddy was president."
The elder Bush, who was defeated by Clinton in 1992, responded by threatening to go public with his true feelings about the president "as a human being and as a person." If Clinton doesn't back off, he said, the family might unleash former first lady Barbara Bush.
While the Gore camp has asked Mr. Clinton to pipe down, the Gore campaign has not refrained from bashing the Texas governor in recent weeks.
"He has to send his father out to defend him," spokesman Chris Lehane told reporters. "People are starting to wonder, does he have what it takes?"
For the remainer of the week, most likely, the White House will signal a cease fire. After that, anything may happen.