The Israeli coach has led the Blues to their first Champions League final and could clinch the Premier League title on Sunday with victory over Bolton, all in his first season.
But that hasn't quelled rumours that club owner Roman Abramovich is searching for a high-profile replacement, with Frank Rijkaard and Sven-Goran Eriksson the latest candidates touted by the British media.
"With big coaches, you give him more credit but you judge him by his results and this is the way I want people to judge me,'' Grant said.
"The doubt was reasonable, but since then we have succeeded. We took more points than any other team since I was here.''
Maybe Grant is still suffering from comparisons with the man he replaced in September, the charismatic Jose Mourinho.
Mourinho's departure was acrimonious, and left fans distraught at losing the coach who had led the club to its first league title in 50 years and then retained it.
Grant was switched from his position as director of football, despite having no Champions League or Premier League experience, prompting incredulity among fans and an assumption that it was a stopgap appointment.
"I understand the doubt in the beginning,'' Grant said.
"It was very reasonable.''
But since losing his first game 2-0 at Manchester United, Grant has lost just one domestic match and, if United fails to win at Wigan on Sunday, could clinch the title.
"I'm not surprised because I'm with the team every day,'' Grant said.
"No one believed and no one expected us to come to this stage but we believed all the time.''
Grant, who refused to say whether Abramovich had been in touch in the lead-up to the Bolton match, attributes his success to applying just the right amount of tactical discipline to a group of talented stars trying to express themselves.
That has led to accusations of boring soccer and a lack of goals that could cost Chelsea its shot at glory.
If the teams finish tied on points, the league title will be decided on goal difference.
United leads its rival by 17 on that count.
One eye on Moscow
But Grant is trying to concentrate only on Sunday's match at Stamford Bridge.
"You have to respect the past, but I'm looking to the present and the future,'' Grant said.
Whether Grant has a place in Chelsea's future could depend on the next two matches, against Bolton and then against Manchester United in the May 21 Champions League final in Moscow.
And Grant admitted that star striker Didier Drogba has yet to tell him whether he wants to stay with the club past this season.
Drogba has been linked with FC Barcelona, Inter Milan and AC Milan, and is rumoured to be heading to wherever Mourinho next ends up.
"Most important for me is what the player says to me and not what I read in the papers,'' Grant said.
"According to the papers, we have bought 300 players and Didier is at six or seven clubs.
"But we have 27 players in the squad. You think every one of them says to me, 'Avram, I am staying.'"