Mourinho blamed the officials for West Ham's first and third goals, saying the opener should have been flagged offside and the move that led to the Arnautovic goal included a foul on Rashford.
United's 10 points from seven games is the same as in 2013-14 under David Moyes, but an inferior goal difference means this is their worst top-flight start since they picked up seven points in 1989-90.
As for Mourinho, Arnautović added: "I have nothing to say".
"I see so-so but in the little two sessions of training we had (since Saturday) everything (was) normal, desire to work and play". In football every day you have to prove why you are wearing the shirt, especially in the big clubs like Manchester United.
Daniel Farke's side sit one point and four places above Derby - who are ninth - in the Championship table and their success comes as no surprise to Lampard.
Asked if some of his players do not care, Mourinho said: "I think that some care more than others".
The full-back Luke Shaw said as much after the damaging defeat at the London Stadium at the weekend, and rather than criticising the player for his verdict of "horrendous" on the performance Mourinho said he felt the same way.
'This game was really bad but everyone has a responsibility for that.
"It's always hard when you lose games and especially when you play for a big club that wants to win trophies".
As Paul Pogba sauntered not unhappily - actually, as he sauntered smilingly - towards the poshest team bus you will ever see, the reporters in what Mourinho sneeringly calls "the famous mixed zone" called out to ask if he would comment.
"I prefer the doctor to speak, but they are not in conditions". Everybody in the club has a role to play, from the kit man and the nutritionist to the manager.
"Luke Shaw for example said something I agree (with), but don't agree totally (with) because I am part of it and other persons are part of it: "We players on the pitch we have to perform, we have to give more". It's hard to take and we're sorry to the fans for what they saw.