"Four years after the bones of Richard III were discovered beneath a car park in Leicester, a survey of a similar site in Reading is now being carried out as part of the Berkshire town's Hidden Abbey Project." Source: Sunday Express.
"Experts using specialist equipment are working on a car park of a prison which they believe was built within the grounds of Reading Abbey."
"They plan to use ground-penetrating radar (GPR) to locate the original boundaries of the Abbey Church and will then scour the area for possible sites of archaeological interest for future investigation, including the High Altar where Henry was buried, the ambulatory and the Lady Chapel."
The remains of Henry I could be under the modern parking lot or a playground in the same area. According to Live Science, "... British historians and archaeologists are turning to a church and school yard in the town of Reading in search of the remains of Henry I, who ruled England from 1100 to 1135. The modern buildings are on the site of the old Reading Abbey, which was shut down — its abbot was hanged for treason — in 1539."
According to Wikipedia, "[t]he abbey was largely destroyed in 1538 during Henry VIII's Dissolution of the Monasteries. The last abbot, Hugh Cook Faringdon, was subsequently tried and convicted of high treason and hanged, drawn and quartered in front of the Abbey Church. After this, the buildings of the abbey were extensively robbed, with lead, glass and facing stones removed for reuse elsewhere."
Abbot Faringdon apparently supported Henry VIII in his attempt to divorce his first wife, Catherine of Aragon, and marry Anne Boleyn. From what I've read, Faringdon was also willing to conform to the king's decision to assume control of the church in England, but he may have been dissembling. Whatever the case, the abbot was accused of providing financial support to the rebellion by Roman Catholics in the north and put to death. It's ironic that the looting of the Abbey Church which followed his grisly execution may have led to the loss of the burial site of the first King Henry.
Related: King Henry I (1068-1135)