Boss Sam Allardyce's need for some extra firepower was illustrated in the Capital One Cup win over League Two Cheltenham on Tuesday night.
The Hammers created a host of chances after Cheltenham's Matt Richards halved the deficit from the penalty spot, but they could not kill the game off.
Assistant boss Neil McDonald admitted: "We'd have been happier if we had scored more goals, from the final 15 minutes of the first half we totally dominated the game.
"We really should have won by a bigger margin and that was the only disappointment.
"We are pleased that everyone had a tough test because Cheltenham weren't easy to play against so we are pleased to go through.
"The majority got 90 minutes which they needed because they haven't been involved and it should have been an easier game, so we have lots to work on."
The problem for Allardyce, as joint-chairman David Sullivan recently stated, is that due to the new wage limit he must move a player out before he can bring another in.
With Andy Carroll sidelined by a heel injury, Modibo Maiga has led the attack in their first two Premier League matches but has rarely looked like finding the net.
Press Association Sport understands Honduras forward Roger Rojas is due to sign on loan, subject to obtaining a work permit, but he is unlikely to be an immediate solution.
Allardyce made 11 changes to his side for the cup clash and at least half of the players who came in - with the exception of Joe Cole, Morrison and new signings Adrian and Razvan Rat - could be expendable.
Vaz Te started up front and although he scored a fine free-kick he is more effective on the wing, where Allardyce now has Cole, Matt Jarvis and Stewart Downing vying for two positions.
The Portuguese attacker, who hit the goal which secured West Ham's promotion in the 2011/12 play-off final against Blackpool, has been linked with Cardiff and Nottingham Forest.
Matt Taylor and Jack Collison did not even make the bench against Newcastle on Saturday and were also in the shop window against Cheltenham.
Joint-chairman David Gold wrote in his programme notes before the match: "I know David Sullivan and Sam will keep working up until the last minute and I wouldn't be surprised at all if we brought in at least one more signing."
Allardyce certainly has a stronger squad than this time last year and Morrison's emergence following a season on loan at Birmingham is another boost.
McDonald added: "The manager stated he is happier with his squad than he has been for quite a while and we have lots of competition for places which is good for West Ham.
"If someone comes in and someone goes out we will let you know."
Midfielder Alou Diarra needs a scan on his knee after departing on a stretcher half an hour into Tuesday night's game.
- Sports & Recreation
- Sam Allardyce
- Ravel Morrison