Fujimori, president from 1990 to 2000, is serving a 25-year sentence for human rights abuses and corruption in special housing set up at a police headquarters in a suburb of Lima.
Now 78, Fujimori has suffered a series of health setbacks that has seen him in and out of hospital.
His adult children -- especially Keiko Fujimori, who currently heads Fuerza Popular (Popular Force), the party that has a lock on the single-chamber legislature -- have long campaigned to get him released.
On paper at least, both the "Fujimoristas" and President Pedro Pablo Kuczynski's Peruvians for Change party share a center-right ideology.
However, strong opposition from Fujimori supporters has thwarted many of Kuczynski's government plans, and has even toppled three of his cabinet ministers.
The latest to bite the dust was finance minister Alfredo Thorne, forced to resign on Wednesday after a no-confidence vote.
Kuczynski, a former World Bank economist, Wall Street banker and ex-finance minister, said Thursday that it is now time "to look at" a pardon for Fujimori.
Peruvian law allows the president to offer pardons for Independence Day -- celebrated here on July 28 -- and for Christmas.
Kuczynski said his government has "always been evaluating" a pardon for Fujimori, and now those talks are "advancing."
Kuczynski cornered
The bad blood between the two leading politicians goes back to the 2016 election, when Keiko Fujimori and Kuczynski both ran for president.
Keiko Fujimori won a majority of votes in the first round of balloting and her party won control of Congress.
But she lost narrowly to Kuczynski in the second round when the smaller left and right-wing parties coalesced in an anti-Fujimori vote.
Keiko had also failed to win a runoff vote in the 2011 presidential election.
Alberto Fujimori remains a controversial figure in Peru, and memories of his corruption-riddled, iron-fisted rule -- and his successful crackdown on the country's two leftist guerrilla movements -- are still fresh.
Kuczynski initially opposed pardoning Fujimori -- and now that he's considering the move, the parties that helped pushed him into the presidency are upset.
"They're cornering him and instead of going out to fight like a statesman, he's caving to pressure from the Fujimoristas," said Geronimo Lopez, head of the CGTP, Peru's most powerful labor union.
No quid pro quo
"The only thing that I want to say is that there is no connection between a humanitarian pardon and Peruvian politics," Kuczynski said.
"Politics is one thing, and the health of a person is another," said Kuczynski, who like Fujimori senior is 78.
The ex-president's youngest son, Congressman Kenji Fujimori, thanked Kuczynski on Twitter.
"It's time to turn the page. I will be eternally grateful," he wrote. Kenji is one of the few Fujimoristas that is in contact with the Kuczynski administration.
Just last week, Keiko Fujimori failed in a legal attempt to get her father released from prison.
She pressed Kuczynski in a Twitter message: "You have been evaluating the case for months, now the time to act has come. Use your authority as president. Pardon him."
Polls show that nearly 60 percent of Peruvians support a humanitarian pardon.
The ex-leader's health has deteriorated in the past three years, with hospital visits to treat ailments that include hypertension and recurring problems with a cancerous tongue lesion.
In 2013 the government of President Ollanta Humala said that Fujimori did not qualify for a humanitarian pardon.
Kuczynski earlier said he supported home arrest for the ex-leader, but Fujimori's supporters in Congress rejected the proposal, insisting on a pardon.
Fujimori was jailed in 2007 for his role in killings by a death squad targeting supposed guerrillas in the 1990s. He was also convicted of embezzlement and bribery.
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