The 50-year-old victim was walking in Captain's Road in Edinburgh at 00:15 when he was hit on the back of the head by an unknown object.
He fell to the ground and lost consciousness before waking to find his wallet was gone.
Officers believe the same man may have been responsible for an attempted robbery a short time earlier.
Paramedics were called to the scene of the second attack and helped the victim, who had suffered serious head injuries.
He was then taken to the city's Royal Infirmary, where he remained in a stable condition on Saturday afternoon.
Police said there was a "strong possibility" that the same suspect was also responsible for an attempted robbery at about 23:30 on Friday night - 45 minutes before the Captain's Road attack.
In that incident, a 17-year-old male was walking home on Gracemount Drive when a man approached him with a red metallic baseball bat and demanded that he hand over money.
The teenager managed to run away without handing over any money or any of his property.
The description given to police in both incidents was very similar, and officers said they were keen to hear from anyone who was in the area who noticed any unusual behaviour.
The male suspect was described as having a mixed race appearance, in his early 20s, about 5ft 6in tall with a very thin build. He spoke with a Scottish accent and had curly stubble on his neck but not his face.
He was wearing light grey jogging bottoms, a hooded top with the hood up and carrying a red metallic baseball bat that measured around 2ft in length and had dark/purplish marks near its handle.
Det Con Alastair Paisley said: "Given the close proximity of these incidents and the time of night they happened, we believe there is a strong possibility that these attacks were carried out by the same perpetrator.
"As part of our ongoing investigation officers are currently reviewing local CCTV and carrying out door-to-door enquires
"We also have a detailed description of the male which, of course, includes a very distinctive baseball bat".
Police are hunting a robber who knocked his victim unconscious before stealing his wallet.