Bayern confirmed today the attacking midfielder would join them on July 1 after they agreed to meet the release clause in his contract.
The timing, on the eve of Dortmund's Champions League semi-final with Real Madrid, perhaps their biggest fixture this century, has not gone down well with club chiefs.
"It could have been timed better," Klopp told a press conference.
"I was told last night at 10pm that it had come out. I've known about it since a day after the Malaga game (the Champions League quarter-final on April 10).
"Now I can say that time heals everything. It's going to take a while, this news is not good for us. The timing of it is not ideal.
"Anybody can make their own minds up as to why it's come out now, but it is out."
Gotze quickly came in for fierce criticism from irate Dortmund fans, who posted messages on his official Facebook page, while Lothar Matthaus, who won seven Bundesliga titles with Bayern, claimed they had every right to "feel betrayed".
According to reports, Dortmund requested a police presence at their training session this afternoon for fear of reprisals.
"The fans feel betrayed. The comments Mario Gotze made in the recent past (about wanting to stay at Dortmund) don't make him credible. This damages the whole of football," Matthaus said.
"Anybody with mischievous thoughts will now think back to the (Champions League) quarter-final against Malaga.
"Gotze was a decisive player and wasted six or seven big chances. He showed nerves then and this whole thing has certainly been developing for a while.
"Maybe now it's just a relief for him that it's out in the open."
On the timing of the announcement Matthaus said: "It's not fair, it's unbelievable."
Bayern's apparent lack of contact with Dortmund over the deal has also seemingly upset the selling club, who earlier revealed Gotze's agent had informed them the 20-year-old wanted to move to the Allianz Arena, but stressed they had heard nothing official from the Bavarians.
In announcing the deal, Bayern claimed they had kept quiet "out of consideration" for Dortmund's clash with Madrid at the Westfalenstadion tomorrow night.
Dortmund pointed out that Gotze acted appropriately at all times.
Gotze was under contract at Dortmund until 2016, with his release clause reportedly set at 37 million euros (£31.5m), meaning he will become the most expensive German player of all time.
Dortmund chief executive Hans-Joachim Watzke said on the club's website: "We are obviously disappointed beyond measure, but emphasise that both Mario and his advisors have behaved within the terms of the contract."
Gotze could face a hostile reception from his own fans tomorrow night, but Klopp wants them to show understanding.
He added: "For everybody who is angry, all I can say is that, a year ago, we signed Marco Reus from (Borussia) Monchengladbach and they hardly broke out in celebration there, but he still played some excellent games for Gladbach. (Reus agreed a pre-contract in January before moving in the summer).
"We will do all we can to ensure our concentration is not disturbed. When you deal with things in life in the right way, then you can draw energy from them."
Klopp claimed the chance to work under former Barcelona coach Pep Guardiola, who has signed a three-year contract to take over at Bayern in the summer, persuaded Gotze to move.
"I've spoken to Mario," he said. "Mario has not taken this decision to spite his team-mates in the slightest. He has not decided to spite the club - he has too much to thank them for.
"He is the player Pep Guardiola really wanted. He didn't want to pass up on the chance to work together with this extraordinary coach."
Losing Gotze, especially to their closest domestic rivals, is a huge blow to Dortmund.
Klopp's side recently relinquished the Bundesliga title they won in each of the past two seasons, with Bayern winning the 'salad bowl' in record time.
Gotze has emerged as one of the most coveted players in European football since coming through the ranks at Dortmund and making his Bundesliga debut in November 2009.
He played a key role as Dortmund won back-to-back Bundesliga crowns and has made 22 appearances for Germany, scoring five goals.