The Swansea-based region have to beat Ulster in their final Pro12 game of the season on Saturday and hope Munster get fewer than two points against Scarlets.
Ospreys have played in the top flight of Europe every season since 2003.
"There are financial implications, but standards come into it as well. We all want to be playing at the highest level of rugby," said King.
"We've got a massive game this weekend, we've given ourselves a lot of work to do, but there still is a mathematical chance that we can do it."
Seventh-placed Ospreys have picked up maximum points from their last three Pro12 matches and are aiming to do the same again against fourth-placed Ulster.
They need to overtake Munster - who are sixth and four points ahead of them - to get into the Champions Cup.
Beating Ulster could do a favour for Ospreys' local rivals Scarlets who are trying to displace the Irish province in the last play-off place.
The Scarlets in turn could help Ospreys by beating Munster at Thomond Park.
King was named man-of-the-match in Ospreys 40-27 win over Cardiff Blues, but is frustrated that the team have left themselves with so much to do.
"We've got 15 points from our last three games it's just a shame that we've left it so late in the season to start playing a bit of rugby and scoring those tries that we've needed," he said.
"There have been a couple of results in the season when we should have won, realistically, and if we'd picked up a couple of points here on there we wouldn't be having to get 20 points in our last four games.
"Ulster have something to play for so they are going to come out all guns blazing so it's going to be a massive test this weekend."
Wales forward James King says Ospreys are desperate to play in next season's European Champions Cup.