His side were well beaten on Saturday with Arsenal cruising to victory courtesy of two Robin van Persie goals and one from Theo Walcott.
At the final whistle, the Israeli shook hands with his players as they left the pitch before applauding the fans and throwing his claret and blue scarf into the stands in what appeared to be a farewell gesture.
However, Grant maintained: "I have done this at the last few games, thrown my scarf to the children who are there.
"Today it was emotional to see the reaction of the supporters after the game, we had lost 3-0 and they were so supportive.
"It was touching to see this, so I did what I did. You need to respect the supporters, that is what I did."
The club are officially making no comment on Grant's position and the 55-year-old was also evasive when it came to the rumours surrounding his future, adding he was the wrong person to ask.
"Those rumours on the day of the game, and it does not matter where they have come from, do not help but I want to focus on the thing I can control and that is the team," said Grant.
"Of course I prefer to answer football questions, I am not the man you need to ask, I am prepared to answer questions about football. Not about things around it."
He added at the post-match news conference: "You know who gave you the rumours. Ask the person, do not ask me."
Grant has led West Ham into the semi-finals of the Carling Cup, where they beat Birmingham in midweek to move within 90 minutes of Wembley.
He was appointed by new owners David Gold and David Sullivan in the summer and was reportedly given a three-game win ultimatum by them over Christmas - although this was later denied.
Grant had a public spat with vice-chairman Karren Brady over the collapse of Steve Sidwell's proposed transfer and that appeared to bring matters to a head.
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