One woman is seen lying on the ground, with much blood around her, as someone presses a towel to her back.
"The victims have been transported to a hospital and are undergoing treatment", the statement said.
A security source told the Reuters news agency that two of the tourists were in a critical condition but did not elaborate.
Spanish media initially reported that three of the people stabbed were Spanish women but later reports suggested they could be Mexican.
He did not specify the nationality of the attacker or of those wounded, nor give details on the severity of their injuries.
Jerash is the second-most popular tourist destination in Jordan, as it contains a large number of ancient ruins and represents an archaeological interest.
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Police have arrested a Jordan-born Palestinian man from a refugee camp in Jerash, identified as 22-year-old Mustafa Abu Touameh.
It is not the first time tourist sites have been targeted by attacks in Jordan. A man sat nearby with an apparent wound to his leg. Please, help him now!'
In 2005, triple hotel attacks killed at least 23 people, while the following year a British tourist was killed when a gunman opened fire at Roman ruins in Amman.
Such incidents are rare in Jordan, with the kingdom considered one of the Middle East's safest tourist destinations.
A 2016 attack by the Islamic State group killed 14 people, including a Canadian tourist.
Jordan has seen an increase in tourists from overseas in recent years and such attacks are rare.