The Belgian was taken to hospital on Friday after collapsing during a reserve-team training session. The 40-year-old had been suffering from a virus and had already missed several training sessions.
Danks visited Kompany on Monday and offered an encouraging update on his condition.
"I paid him a short visit yesterday," Danks told reporters ahead of Bayern's German Cup second-round clash against Magdeburg on Wednesday.
"He is in good spirits. It looks like every day he is getting better. There are no major updates in terms of diagnosis or return to work. Let him rest, let him take his time."
Bayern have a demanding schedule ahead, with Wednesday's cup tie followed by Saturday's Bundesliga showdown against Borussia Dortmund. The German champions will then travel to Spain next week to face Atletico Madrid in the Champions League.
Despite remaining in contact with Kompany, Danks said their conversations had focused on his health rather than football.
"We had some good conversations with him but kept it away from football and specific tactics," he said. "It was just a friendly chat and nice to see him and catch up."
Danks also reflected on an emotional week for the Bayern squad, which has celebrated Harry Kane's Ballon d'Or triumph while dealing with concern over Kompany's health.
Kane became the first Englishman in 25 years to win the prestigious award, following Michael Owen's success in 2001.
Danks said the contrast between the celebrations and Kompany's health scare had tested the squad emotionally, but its unity had helped the players remain focused.
"For us, we are super professional. Full extremes of life in the last week. Joys and highs for Harry and the lows of the last days," he said.
"To get through these moments we had to rely on our chemistry, our team spirit and that definitely helped us."