According to the WHO, tap water is safe for consumption for 98% of the country.
Tap water in Jordan is generally safe to drink, but for a short trip it's better to stick to bottled water. This is readily available but check the seal has not been broken. Alternatively, you can boil tap water for 10 minutes, use water purification tablets or a filter.
The tap water in southern Jordan, particularly Wadi Rum, comes from natural springs at Diseh and so is extremely pure. Avoid drinking water from wadis in the wild as pools may have been used as waterholes for livestock. In the Jordan Valley, amoebic dysentery can be a problem.
If you get stuck in the desert without water, remember that you are more likely to be seriously ill (and even die) from dehydration than you are from an upset stomach, however unpleasant it may be. In summary, if water is offered and you need it, worry about its provenance later!
According to the CDC tap water is generally not safe for consumption.
According to the Numbeo water drinkability and accessibility index, tap water has a score of 46.15 / 100 (100 being best).
Numbeo considers this score to be "Moderate"
1 adult could save about 431$ per year in Jordan by drinking tap water instead of buying bottled water.
๐ณ Drinking tap water would also help with reducing your plastic footprint and improve environmental pollution.
Like in most countries, water quality and accessibility differs from location to location, therefore it's the best to check the information available on that place.