"Fourteen Jordanian pilgrims died and 17 others were missing" during the performance of hajj rituals, Jordan's foreign ministry said in a statement.
The ministry later confirmed the 14 had died "after suffering sun stroke due to the extreme heat wave".
Iranian Red Crescent chief Pirhossein Koolivand separately said "five Iranian pilgrims have lost their lives so far in Mecca and Medina during the hajj this year", without saying how they died.
The hajj, one of the world's biggest religious gatherings, is one of the five pillars of Islam and all Muslims with the means must perform it at least once.
Temperatures have pushed well past 40 degrees Celsius (104 degrees Fahrenheit) during the annual pilgrimage that around 1.8 million Muslims are taking part in this year.
Many of the rituals are performed outdoors and on foot, creating challenges especially among the elderly.
Saudi Arabia has not provided any information on fatalities.
But the kingdom has implemented heat mitigation measures, including climate-controlled areas. It distributes water, and offers advice to pilgrims on protecting themselves from the sun.
During last year's hajj at least 240 people -- many from Indonesia -- died, according to figures announced by various countries which also did not specify causes of death.
More than 10,000 heat-related illnesses were recorded last year, 10% of them heat stroke, a Saudi official told AFP this week.
A Saudi study said regional temperatures were rising 0.4 C each decade, and worsening heat may be outpacing mitigation measures.