Several names have been linked with moves to Merseyside in recent days as Everton look to build on the progress made since Moyes' return.
One recurring theme has been the search for defensive reinforcements.
West Ham defender Aaron Wan-Bissaka has emerged as a player of interest, with reports suggesting Moyes views the former England international as a potential solution to the club's right-back concerns.
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The departure of veteran Seamus Coleman has increased the need for competition in that position, while Jake O'Brien spent large parts of last season operating out of position on the right side of defence.
Elsewhere, Everton have been linked with Man City centre-back Nathan Ake.
Reports suggest the Netherlands international could be available for around £20 million as City continue reshaping their squad. Ake's Premier League experience and versatility would make him an attractive option should Everton pursue the move further.
The Toffees are also understood to be monitoring free-agent opportunities, with former Aston Villa and Newcastle United left-back Matt Targett among the players linked with a switch to Merseyside.
However, one of Everton's most ambitious pursuits appears to have hit an early obstacle.
Everton get Middlesbrough reply after opening Hayden Hackney talks
Earlier this week, David Ornstein reported that Everton opened talks with Middlesbrough over signing Championship Player of the Year, Hayden Hackney.
However, according to Sky Sports News reporters Lyall Thomas and Vinny O'Connor, Middlesbrough have rejected Everton's initial approach for the 23-year-old.
The report states that Everton remain some distance away from Boro's valuation of the player and that no further discussions are currently taking place between the two clubs.
That does not necessarily mean the transfer is over.
Hackney is one of the Championship's best midfielders and Boro were always expected to demand a significant fee for a player viewed as one of their most valuable assets.
Everton's opening enquiry may simply have been designed to establish the parameters of a potential deal before deciding whether to return with an improved proposal.
Competition could also complicate matters.
Sky Sports report that Tottenham, Nottingham Forest and Crystal Palace are among the other clubs tracking the Middlesbrough academy graduate.
Interest from multiple Premier League sides naturally strengthens Boro's negotiating position and reduces the likelihood of any discount.
Hackney has been attracting admirers for some time.
The midfielder has become a key figure at Middlesbrough over the last few seasons, developing into one of their standout performers.
His technical ability, passing range and composure in possession have consistently earned praise, while his age profile makes him an attractive target for clubs planning both for the present and the future.
Moyes has spoken previously about the importance of improving the quality and depth of his midfield options, particularly as the club aims to continue climbing the Premier League table.
Hackney's energy and creativity could offer something different to Everton's existing midfield unit.
The challenge, however, may be financial.
Sky Sports' update suggests Middlesbrough's valuation may be considerably higher than Everton had hoped. Unless there is movement from one side or the other, negotiations may remain stalled for the time being.
That said, transfer windows often change quickly.
Everton have shown throughout the summer that they are willing to explore multiple options across the market, and Hackney's name remains firmly on their radar despite the current impasse.
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