The Norwegian was appointed Cardiff manager in January, but his arrival has failed to produce an upturn in fortunes and the Welsh outfit have won just twice in the Premier League during his tenure.
Solskjaer was left particularly disappointed following a 2-1 defeat at Everton on Saturday, with Seamus Coleman grabbing a last-gasp winner.
The result leaves Cardiff second-bottom of the table and three points away from safety with just eight games remaining.
Solskjaer is confident his side can avoid the drop, but the former Manchester United striker will not turn his back on the club if they make an immediate return to the Championship.
"We each want to keep our place in the Premier League, but no matter what happens I will be here for the future. I can promise to be 100 per cent loyal," he said.
"Everybody knows that continuity is the key factor to success. I sensed a great buzz for Wales from the start. The people are fantastic.
"The passion in this part of Wales for Cardiff City to keep their place in the Premier League seems to be confirmed day after day.
"Commitment, loyalty and hard work are values I believe are important. I have never worked as hard in my whole career, but I knew that when I took the job.
"I want to earn respect through the football my team plays and commitment to the club. This is a major project and I am committed to it.
"Right at the moment, though, my whole focus is on doing everything possible to ensure Cardiff City are still in the Premier League next season.
"We have big challenges ahead, but we are ready for those. Nobody at Cardiff City doubts we can keep our place in the Premier League. We have eight cup finals in which to achieve that."
