The 65-year-old woman was walking on Mansfield Road in Musselburgh, East Lothian, when she was approached by the two teenage boys.
One of the youths made a grab for her bag, but the pensioner refused to let go and struggled with the mugger.
After a short time, the suspects gave up and left empty-handed, in the direction of Dalrymple Loan.
The first suspect is white, around 16 years old, and wore a blue jumper. The second suspect is also white and around 16.
A police spokeswoman said: "The female in question was able to resist the attempted robbery and, whilst fortunately uninjured, she has been left extremely shaken by the incident.
"We are eager to locate those responsible, and are therefore seeking information from any member of the public who was in or close to the vicinity of the incident at the time to come forward with information. Similarly, anybody who recognises the descriptions of the suspectsis also encouraged to come forward."
If you have information call police on 0131-311 3131, or Crimestoppers on 0800-555 111.