Moyes is seeking to strengthen his West Ham side this month as they prepare for a fight to stay in the top-flight.
Neither manager seemed overly impressed by their team's start to the half, and both sides opted for a change after 10 minutes, Everton bringing on Theo Walcott for Djibril Sidibe, while Arthur Masuaku replaced Pablo Fornals for the hosts.
Goalkeeper Darren Randolph will certainly make his 2nd launching for the club after rejoining from Middlesbrough, with Lukasz Fabianski out for a fortnight.
After a bright start to the current campaign, however, West Ham's form nosedived and immediately after defeat at the end of December by Leicester City at the London Stadium which left them in the bottom three, Pellegrini was sacked.
If Barkley does decide to leave Chelsea, West Ham would be a good move for him.
Dominic Calvert-Lewin's ninth Premier League goal of the season secured the Merseyside club a point in the capital, the 22-year-old firing home on the stroke of half-time after Issa Diop had headed West Ham into the lead from Robert Snodgrass' free-kick just four minutes earlier.
Moyes returned to the London Stadium in December for his second spell in charge of the Hammers, and the Scotsman made it clear that the squad needs some fresh faces to help them pull away from the relegation dogfight.
"We still made Jordan Pickford make some good saves but I still believe we should have made more of the opportunities we had".
"I think the football manager can ease any of the burdens on the owners by their team's performance. I don't want to lose him, I want to add more of him to the club".
"There's certain circumstances why Ross hasn't featured as much this year but I've got a lot of faith in Ross".