The person said Gibson will be charged with misdemeanor battery as part of the deal. The actor-director's attorney, Blair Berk, said Gibson sought the plea agreement after prioritizing the interests of his children, including his 1-year-old daughter with Grigorieva.
"I know from almost 20 years as a criminal defense lawyer that sometimes justice can come for a client at too high a personal price," Berk said in a statement. "That is particularly so for Mel, whose right to due process can only be exercised in this case with an enormous media circus attached."
Grigorieva claims the 55-year-old actor punched her in the face, breaking her teeth, during a dispute at his Malibu home in January 2010. She said he was also verbally abusive.
Gibson admitted in court documents only to slapping the Russian musician to restrain her from shaking their baby, Lucia.
The former couple ended their 2-year relationship last April and has been locked in a bitter custody dispute over their daughter for months.
Grigorieva's accusations came out in audio recordings leaked last year during the custody battle. The recordings also contained racist and sexist remarks. Gibson has never addressed the recordings.
The actor was expected to be formally charged with battery this week, and the plea deal may allow Gibson to avoid jail time.
"It's good news in that he basically enters his plea, he does some domestic violence counseling and it's over," said ABC News legal analyst Dana Cole. "It's bad news in that he's chalked up a criminal conviction on his record, even though that can be expunged over time."
Cole said aside from undergoing anger management, Gibson will likely be placed on probation.
"So if he trips again or if he has an act of rage or violence against women that can land him in jail, so he's going to have to be on his best behavior for the next couple, three years," he said.
Prosecutors have not yet announced a decision on whether to charge Grigorieva with extortion, which the Academy Award winner has been seeking. Gibson said she tried to extort millions of dollars from him by threatening to release phone recordings after their relationship ended. The L.A. County Sheriff's Department finished their investigation into the allegations last month.
If charged and convicted with extortion, Grigorieva could face up to several years in prison. However, experts say the case would be difficult to prove since it's common to use incriminating evidence as a bargaining chip in civil law cases.
The Associated Press contributed to this story.