On Friday, Mourinho dubbed the Frenchman "a specialist in failure" for going almost nine years without winning a trophy.
Wenger was then on solid ground to offer a withering response after Arsenal edged a rousing FA Cup fifth-round tie against Liverpool 2-1 at the Emirates Stadium.
"I don't want to get into those silly disrespectful remarks," Wenger told BT Sport television.
"I never spoke about him in my press conference and I will not start that. The only thing I know, it's more embarrassing for Chelsea than for me.
"I am embarrassed for him ... honestly."
Twenty-four hours after Mourinho's verbal attack on Wenger, his Chelsea side lost 2-0 at Manchester City in their fifth-round encounter.
"I am more disappointed for Chelsea than for me," added Wenger of the Portuguese's verbal attack. "I am not interested in the subject at all and that is really genuine.
"Honestly I cannot force my interest in things I am not interested in," he told reporters. "I love football and I am sorry to disappoint you but what is interesting for me is what happens on the pitch."
Mourinho's criticism was in response to Wenger who had said earlier on Friday that the Premier League title was Chelsea's to lose as they were top of the table.
FEAR OF FAILURE
He also hinted that the Portuguese had ruled his side out of the race because of a fear of failure.
In terms of matters on the pitch, Arsenal were not at their best against Liverpool but took a 2-0 lead with goals from Alex Oxlade-Chamberlain and Lukas Podolski.
Liverpool struck back with a Steven Gerrard penalty after a foul on Luis Suarez, missed several other good chances and were denied a second spot kick by 2010 World Cup referee Howard Webb.
Suarez, tripped by Podolski for the first penalty, was involved again when he was clattered by Oxlade-Chamberlain but this time his appeals were waved away.
"The second one was even clearer than the first," said Liverpool manager Brendan Rodgers.
"Howard was in a great position. I am not sure if it was because it was so close to the other but we are bitterly disappointed.
"It looked a blatant penalty. It can be the defining moment once again."
The same referee also denied Liverpool a penalty when Suarez appeared to be fouled by Samuel Eto'o in their 2-1 league defeat at Chelsea in December.
Rodgers said he hoped Suarez's reputation for going to ground too easily did not cloud Webb's judgement.
"I think it's the type of player he is. We provoke a lot of challenges because of the aggression in our offensive game," the Liverpool manager explained.
"Luis is always in the box and quite a lot of the time he does not get what he deserves. The one at Chelsea was a clear penalty and if it is a penalty he should have had it independently of his reputation."