The attack took place in a central area of the Ukraine capital, with pictures of blood splattered pavements quickly circulating on social media sites.
Press Association Sport understands the three people concerned have now left hospital and have opted not to co-operate with police investigations.
Around 2,000 England fans are expected to attend Tuesday night's game though and police have reported a relaxed atmosphere in a city where there were no problems last year despite a much larger number of supporters for two games in Euro 2012.
"We are aware of what happened last night and we will be going round most of the major bars, including this particular one to make sure there are no problems," said Rachel Barber, head of the UK police delegation in the Ukraine on Monday.
"We would urge supporters to enjoy themselves but to do so without upsetting anybody.
"I am pleased to say that currently, the atmosphere in the town is good, so in that sense what happened last night was out of the blue.
"Sometimes there can be an element of bravado on social media and you can never legislate for something happening but hopefully the England fans will just have a good time in what is a fantastic city."