Solskjaer was appointed as manager on January 2, replacing Malky Mackay who departed in controversial circumstances at the end of December.
The former Manchester United striker brought in numerous players during the January transfer window as he looked to put his own seal on the team, but it was not enough to avoid the drop.
Cardiff's 3-0 defeat at Newcastle United last weekend confirmed their relegation, which had looked likely for weeks and although Solskjaer has led the club straight back to the Championship after they earned promotion last season, he is sure he retains Tan's trust.
"We did have a good chat last night (Saturday)," the Norwegian said after Sunday's 2-1 home defeat to Chelsea rounded off their campaign.
"I know where I want to go and what we have to do as a club. It's important for me that he trusts me to keep managing the club and taking it forward."
Following what has been a disappointing campaign, Solskjaer is ready to wield the axe and get rid of those whom he does not believe have the right attitude for the Championship.
"There will be changes," he added.
"It's got to be lads who love playing football (who stay at the club), love that side of it, not the other side of football - the cars and all that flash bits.
"The players here have got the experience of the Championship and that experience is something we've got to draw on.
"We will have to also strengthen some positions and be ready to go on August 9."
