The Guardian writes that Lampard will be in charge of Everton for Friday's FA Cup third-round tie against Manchester United, but his future beyond that is uncertain.
And the former England international could be replaced in the dugout by his ex-team-mate Rooney.
The DC United head coach is one of several managers Everton are monitoring, according to The Sun.
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Then-Derby boss Rooney rejected the chance the interview for the role before Lampard was appointed at the end of January 2022, but he may be more open to an approach this time around.
Everton have also been linked with a swoop for their former manager Roberto Martinez, who is available after leaving the Belgium job after their disappointing World Cup campaign in Qatar.
Everton were thumped 4-1 by Brighton in the midweek round of games (Image credit: Getty)
Sean Dyche and Ange Postecoglu are also under consideration by the Goodison hierarchy, but Lampard could yet earn a stay of execution.
Upcoming Premier League games against fellow strugglers Southampton and West Ham will be crucial to determining his future, although a poor result against United on Friday could spell the end for the 44-year-old.
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Greg Lea is a freelance football journalist who's filled in wherever FourFourTwo needs him since 2014. He became a Crystal Palace fan after watching a 1-0 loss to Port Vale in 1998, and once got on the scoresheet in a primary school game against Wilfried Zaha's Whitehorse Manor (an own goal in an 8-0 defeat).
