Grant insists he has the backing of the club's owners as he prepares for the vital London derby against Arsenal and says his team - who remain rock bottom of the table, yet have lost just one of their last seven matches - can stay up if everyone at Upton Park accepts the reality of their relegation struggle.
Grant said: "I believe very much in this club. But I believe we need to look in the mirror and see what we are now. We were not Barcelona in the last year and the years before.
"I don't say anything against the people in the past but our target is very clear, not to be relegated.
"I'm sure there is a very good future here at West Ham if we do the right things."
Grant, who has been under intense pressure over the past two months with Martin O'Neill linked with the job at Upton Park, says he retains a strong working relationship with the club's joint owners, David Gold and David Sullivan.
Nevertheless, failure to take anything out of the next two tough games against Arsenal and Everton could see him jettisoned before the second leg of the League Cup semi-final against Birmingham.
Grant said: "We were speaking about the situation and David Gold was very supportive, in the last game David Sullivan and I were speaking about players to make the squad stronger. We are busy about this."
Wayne Bridge is likely to make his debut at left-back against Arsenal after his loan signing from Manchester City.
And Grant is confident of adding two or three more players to his squad before the January transfer window closes.
Meanwhile, Arsene Wenger accepts football is very much a confidence game - but has every belief Arsenal can deliver the required performance to bounce back from their midweek Carling Cup horror show with victory at Upton Park.
Chelsea midfielder Frank Lampard admits the Blues have not shown the same drive on the field as last season, while new Liverpool boss Kenny Dalglish acknowledges his squad are certainly in need of a lift of self-esteem following another disappointing result at Blackpool.
Wenger openly admits the Gunners were well below par as they went down 1-0 at npower Championship strugglers Ipswich on Wednesday night, despite having selected a strong side.
And while Arsenal still have the second leg at Emirates Stadium in a fortnight to turn the tie around and stay on course for Wembley, there will little room for error at Upton Park if they are to keep up with Premier League leaders Manchester United.
"You think we are all arrogant, confident people, but when you do not have success for a while, the doubt creeps in quickly. It is about belief," Wenger said.
"Kenny Dalglish has been in football all his life, and if he speaks about that, it is really the truth."
Wenger continued: "We have belief, yes.
"We are on a good run in the Premier League, we believe in our strengths, that is why it is always important to respond strongly to disappointments."
The Gunners could reduce the gap at the top down to two points again, as United are not in action until Sunday afternoon when they travel to Tottenham.
Wenger will not settle for another "off night" at Upton Park.
"We are of course sorry [for defeat at Ipswich] and we want to put things right in the second leg. We had a completely average performance," he said.
"If you play like that at West Ham you will lose. We will not play like that at West Ham and we want to win at West Ham.
"What is important for us is we go to West Ham and expect them to be up for it and for us to be prepared mentally.
"It is a Premier League game and is a derby, this season the Premier League has been very tight in every game and so we expect us to be completely produce a top-level performance to win the game, that is what we want to achieve."
Arsenal will be without goalkeeper Lukasz Fabianski (shoulder) for the trip, with Wojciech Szczesny again deputising.
Centre-back Sebastien Squillaci (hamstring) is also out and right-back Bacary Sagna serves the last of his three-match ban.
Denilson has recovered from a thigh problem, while both Robin van Persie and Samir Nasri return to the squad having been rested for the Ipswich defeat.
Midfielder Abou Diaby (calf), goalkeeper Manuel Almunia (ankle), Abou Diaby (calf) and centre-back Thomas Vermaelen (Achilles) remain out.