Following a haphazard first round of voting in April, which took weeks to produce a concrete result, electoral authorities have counted 98.27 percent of ballots from Sunday’s second round.
Fujimori’s 50.005 percent puts her narrowly ahead of Sanchez, who has 49.995 percent of votes.
On Friday, Sanchez suggested that the pair jointly request a recount of run-off votes, citing alleged irregularities in Lima and voting from abroad.
“I propose that we jointly request an exhaustive review, a recount of the entire process, especially where there are alleged indications that transparency may not have been upheld as it should,” he announced at a press conference.
The request, which would have to be submitted to electoral authorities, also followed a demand from Fujimori’s party to “annul our votes from the south”, Sanchez added.
Fujimori took the lead late on Wednesday after votes from abroad were tallied.
The count has now entered its final phase, with challenged and disputed tally sheets under review.
The standard procedure can take several days – something keenly felt in Peru this time given the first round’s razor-thin results.
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The final tally could take between two and three weeks, according to electoral authorities.
The race sees Fujimori make her fourth presidential bid.