Everton have started the new season unbeaten, but their summer window left David Moyes short of attacking options.
The Toffees beat Crystal Palace 2-0 before defeating Preston North End 4-0 in the EFL Cup.
They then drew 1-1 at Bournemouth, with James Tarkowski scoring a stoppage-time equaliser.
However, Everton ended the window with only Thierno Barry as their recognised senior centre-forward.
Beto joined Fiorentina permanently, while Iliman Ndiaye completed a £65 million move to Manchester City.
Everton also failed to complete a £40 million deal for Monaco striker Folarin Balogun on deadline day.
The club additionally saw a loan approach for Manchester United forward Joshua Zirkzee rejected before the window closed.
According to The Sun, Everton are considering a move for former Premier League Golden Boot winner Jamie Vardy.
The 39-year-old is a free agent after leaving Cremonese at the end of last season.
That status allows Everton to sign him despite the closure of the Premier League transfer window.
Vardy spent last season in Serie A but wants to return to the United Kingdom with his family.
The former Leicester City striker attracted interest from other clubs during the summer.
Wrexham and Sao Paulo were linked with him, although Vardy rejected an approach from Wrexham.
Vardy joined Leicester from Fleetwood in 2012 and later scored 200 goals across 500 appearances.
He also won the Premier League Golden Boot in 2020 during his long spell with Leicester.
Everton could now turn to him after finishing the window with just 18 outfield players available.
Brennan Johnson can operate centrally if required, but Everton currently lack another specialist striker behind Barry.
Moyes strengthened elsewhere, with Jack Grealish arriving on loan and Hayden Hackney joining permanently from Middlesbrough.
Christian Norgaard also arrived from Arsenal, while Merlin Rohl and Tyrique George completed permanent moves.
Vardy therefore represents a possible post-deadline solution as Everton consider adding another senior forward.
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