Rangnick says he urged the club to land an attacker and pointed out that Dusan Vlahovic, who ended up at Juventus, Luis Diaz - now at Liverpool - and incoming Manchester City forward Julian Alvarez were all available.
He told reporters: "The answer at the time was no, there was no player on the market that could really help us. [But] there were a few: Diaz who is now at Liverpool, Alvarez who will be at Manchester City in the summer, Vlahovic who at the time still was with Fiorentina.
"Those are just three of them that come across my mind now," said Rangnick.
"We had four days off at the time and on the Sunday I was informed about the issues with Mason Greenwood and obviously Anthony Martial had already left and then I was aware that within four days we had some strikers missing and it might make sense.
"We were still in three competitions - Champions League, FA Cup and fourth in the league, but that's in the past and it doesn't help us anymore.
"I spoke to the board and said 'shouldn't we at least speak and try and analyse if we could get a player either on loan or as a permanent deal', but in the end the answer was no.
"I still believe that we should have at least tried. If we would've found and been able in 48 hours, 48 hours is short notice but it's still 48 hours. It might have been worth to try and internally discuss but we didn't and it was not done."