Both the Foxes and the Toffees are still searching for their first victories of the new Premier League campaign, leaving them in or just above the relegation zone.
Leicester's movement in the transfer market has been restricted by their issues with FFP and the saga involving Wesley Fofana, which is seemingly close to reaching a conclusion.
Meanwhile, Everton are in the market for creative players having scored just three times in four league fixtures and after giving permission for Dele Alli to move to Besiktas.
According to The Sun, the two clubs are contemplating whether to make a formal approach for Pereira over the coming days.
The Brazilian has spent the last year of his career in Saudi Arabia with Al-Hilal, contributing three goals and 15 assists from 34 appearances.
However, the report claims that the 26-year-old is now ready to return to Europe, despite having four years remaining on his contract.
Pereira initially enhanced his reputation during his time with West Bromwich Albion, chipping in with 20 goals and 26 assists from just 77 outings in all competitions.
As many as 11 goals and six assists came from his 33 outings in the Premier League, a return which has seemingly not gone unnoticed by Leicester and Everton.
Pereira helped Al-Hilal win the Saudi Professional League title in 2021-22 with Ramon Diaz's side edging out Al-Ittihad by two points.
Moussa Marega, Odion Ighalo and Bafetimbi Gomis - formerly of Porto, Manchester United and Swansea City - are also currently at Al-Hilal.
Last summer, the Blue Waves allegedly paid in the region of £16m for Pereira's signature, and they would expect a similar fee should they part ways with one of their key men before Thursday's transfer deadline.
Other Premier League teams may also express an interest if it becomes apparent that Pereira could return to English football.