The 25-year-old has played just once this season for the Reds, who are keen for him to play regular first team football in order to regain match fitness.
Ings was given a 17-minute run-out in the Carabao Cup loss to Leicester City but with Liverpool having crashed out of the competition opportunities are likely to be few and far between for the rest of the campaign.
According to The Sun, new West Ham boss Moyes is an admirer of the England forward having tried to sign him while manager of La Liga side Real Sociedad, but both AFC Bournemouth and Newcastle United are also interested.
The trio will battle to convince Liverpool they can provide Ings with the playing time the Merseysiders desire during the second half of the campaign.
West Ham's need for a new forward has been highlighted in recent days after summer signing Javier Hernandez picked up a hamstring injury while on international duty with Mexico.
Among Moyes' main tasks at West Ham will be address the lack of goals accumulated by the east Londoners, with the club having netted just 11 in as many top flight games.
Former Burnley forward Ings has been dogged by injury ever since moving to Anfield in the summer of 2015 and has twice undergone knee surgery. This season his appearances have been restricted to the Under-23 side in Premier League 2 in which he has made seven outings.
Both Liverpool and Ings know a loan move to a Championship club is a viable option but they want to explore the option of a temporary Premier League deal to ensure he is playing at the summit of the sport.
The Mirror reported in October that Liverpool had agreed that a January loan move for Ings would be the best step to take to ensure he returns to full fitness, but that manager Jurgen Klopp has not given up on the striker playing a future role in the first team.
But speaking prior to those reports, Ings insisted he has no plan to take the easy way out in search of regular first team opportunities and is ready to fight with Roberto Firmino, Sadio Mane and Daniel Sturridge for game time.
"Everywhere I have been, I have always found a way to make it happen. I have been in a lot of tough situations at previous clubs where it has been hard to break in," Ings told The Times.
"People from the outside looking in will probably have the assumption, 'Oh, he's the forgotten man' but it has not crossed my mind at all. I still believe in myself. I still believe I can play in this team.
"When you are at a club like Liverpool, it is not a club that you want to walk away from. It would grate on me for the rest of my career, and the rest of my life, if I had not done everything to give it my all. I still feel like I have a big part to play in this football club."