But another source said Herman hasn't ruled out an offer from Texas, which has been expected to make Herman its top choice if Charlie Strong is fired.
LSU has scheduled a news conference for 1 p.m. ET Saturday.
USA Today Sports reported that Herman, in his second season as Houston's coach, met with LSU officials on Thursday afternoon.
Florida State's Jimbo Fisher was also near the top of LSU's wish list, after being targeted last year when it appeared Les Miles might be fired. But LSU's search over the past few days shifted from Fisher to Herman. A source told ESPN that LSU interim coach Ed Orgeron is the clear-cut "backup plan" should it be unable to hire Herman.
"Contrary to media reports, there has been no decision made on who will be the next football coach at LSU," the school said in a statement released during the 25th-ranked Tigers' 54-39 victory over No. 22 Texas A&M on Thursday.
"Coach Orgeron has done a great job of leading this team since taking over in late September and we are pulling hard for the Tigers tonight against Texas A&M. As we stated earlier, we are going to take our time and make sure we select the right person to lead our football program."
Herman also downplayed reports linking him to other jobs during a brief interview with ESPN before the Cougars' regular-season finale Friday against Memphis.
"I'm not going to talk about any other job except the one I've got right now and these kids," Herman said.
Herman said that he told his players to not "believe anything that you read," emphasizing to ESPN that there is "nothing to handle except beating Memphis."
Houston ended up losing the back-and-forth game to Memphis, 48-44.
A source told ESPN earlier this week that Texas had decided to fire Strong in the wake of a loss to Kansas last Saturday. But Texas athletic director Mike Perrin insisted Strong would be evaluated after the season is over.
Strong told ESPN on Thursday that he has heard nothing different about his status from Perrin or president Greg Fenves and that his sole focus is beating TCU on Friday.
Fenves' preference has been to keep Strong, sources told ESPN's Mark Schlabach, and the odds of Strong returning for a fourth season are expected to increase dramatically if Herman is no longer available. The Longhorns need to beat the Horned Frogs to finish 6-6 and become bowl-eligible.
Cougars athletic director Hunter Yurachek told the Houston Chronicle he was unaware of any changing job prospects for Herman.
"I have not been informed by Tom or his agent that they are in conversations with LSU," Yurachek said via text, according to the Chronicle. "We are in Memphis together, preparing for a game."
A source told ESPN that Yurachek met with Herman's agent, Trace Armstrong, earlier this week and offered a contract extension to the coach that would include a seven-figure raise. Herman makes $2.8 million annually and is the highest-paid coach at a Group of 5 conference school.
As of early Friday morning, Yurachek and Houston athletic department officials have not received a response from Armstrong or Herman on the offer and were unaware of any agreements Herman might have made with another school, but were anticipating his departure for another program within the next 48 hours, the source said.
Billionaire Tilman Fertitta, chairman of the Houston board of regents, said earlier this week he had no concerns about money being a deciding factor to keep Herman.
"I do not fear a bidding war for Tom," Fertitta said. "We're not going to lose him over money."
HornsDigest.com first reported that Herman and LSU were close to a deal.
Orgeron has guided LSU for the bulk of this season after Miles was fired on Sept. 25 following an 18-13 loss to Auburn. The Tigers are 4-2 under Orgeron, who has been a candidate for the full-time job.
Information from ESPN's Sam Khan was used in this report.