Bayern beat Chelsea 3-0 in the first leg of their Champions League last-16 tie at Stamford Bridge.
Cole says former team-mate Frank Lampard would have been more desperate to make January additions after drawing Bayern in the last 16.
“I think knowing this game was on the horizon, Frank would have been desperate to get players through the door," Cole said on BT Sport.
“You've asked Giroud, who's done brilliant for you at the weekend, he's not been fit and a 32-year-old striker to come and lead the line against probably one of the best sides in Europe as the moment, it just looked like he ran out of steam towards the end.
“So not recruiting has definitely affected the performance and the team.
“Bayern Munich were sublime from start to finish."