i kinda look at it and think that if we're signing a player from, say, portugal or argentina or anywhere that isn't basically the UK, they're highly unlikely to know who everton are. they might know we're in the premier league, but beyond that i doubt the vast majority would care about anything other than the fact that we're premier league and that after the initial contact they'll do some fact-finding at that point, same as the rest of us do when we're looking for work. their agent will consider whether we're a good move for their client financially (i.e. how much are we paying and for how long) and professionally (are we at a suitable level, chance of client being more visible as a result of playing for us, profile of the club, likelihood of european football etc), and they'll probably consider whether they'll consider living in and around liverpool.
ultimately, we have a director of football role to identify and recruit suitable players. if i were him, i'd suggest that there's never any guarantee of who a manager will be, how long they'll last, or anything like that these days anyway, and that he's recruiting to improve the squad. he can even put in a release clause if necessary - if the new manager doesn't play you in x% of games where you're fit and available, you can move elsewhere for what we paid for you or with the manager's agreement. plenty of ways around it.