The renewed talks were initially reported by the New York Post.
Earlier discussions on a deal fell through as the companies were unable to agree on a price, Reuters reported in April.
Altera's shares jumped 7 percent to $47.50 in premarket trading on Monday, while Intel's shares were up slightly at $33.15.
One way or the other, an outcome is expected in a few weeks, the newspaper said, citing a person familiar with the matter.
Intel and Altera could not immediately be reached for comment.
Deal talks between the two companies first surfaced in March and media reports suggested that a potential deal could top $10 billion, making it Intel's biggest acquisition ever.
—CNBC's David Faber and Reuters contributed to this report.