A club statement said that Agar would have a future role at Leeds but in what capacity he will remain at the club has yet to be confirmed.
Leeds' only win of the season came against Wakefield Trinity earlier this month, with Rhinos losing five of their six Super League matches so far this campaign.
Jamie Jones-Buchanan has been appointed as interim head coach.
Agar won the Challenge Cup with Leeds in 2020 but despite some strong recruitment ahead of this season, the Rhinos have endured a difficult campaign.
“When I first took on the head coach’s job three years ago, I agreed with Kevin Sinfield and Gary Hetherington that I would do so on the basis that I could help take the team forward," explained Agar.
"I agreed a six month rolling contract and I was happy to continue whilst ever we were making progress. However, I don’t feel like I am having the impact I would hope to on the squad.
“I am proud of our record over the last three years. The club was in a tough position when I took over, we battled against relegation before facing the unprecedented challenges of Covid and an injury-ravaged squad last season.
"On the field, we won the Challenge Cup at Wembley, reached the play offs in the last two seasons and saw the emergence of a number of highly-promising young players.
"I really believe we have a strong squad who have the potential to achieve this year. They are working hard in training every day but unfortunately this has not materialised in to results and ultimately that responsibility rests with me."
Chief executive Gary Hetherington says the club have no timescale when it comes to appointing a new head coach.
He said: “On behalf of everyone at the club, I would like to thank Richard for his outstanding effort as head coach of the club.
"Normally with teams in this position, it is easy to identify the causes of a down turn in form such as this but in our case it is a difficult one to solve. The teams preparation has been good each week, the spirit within the group has been strong throughout and our coaches and backroom staff have worked tirelessly to provide our players with excellent support.
"We started the season with what we believed was a very strong squad with an outstanding crop of talented young players, but we have simply not converted that into team performances.
“We do plan to appoint a new head coach and will apply a comprehensive recruitment process but there is no pre-determined timescale for this appointment.
"This Saturday we entertain Castleford Tigers at Headingley and the Challenge Cup offers another opportunity to get our season started. All our focus will be on this game and then St Helens the following week as we return to Super League action.
"Jamie Jones-Buchanan is excited about his new opportunity and I am sure he will bring the same passion, enthusiasm and expertise to this new role as he showed throughout his playing career."