The Scottish SPCA rescued the pup yesterday evening after it was found on the beach at the bottom of King's Road.
A spokeswoman said the pup will be taken to Middlebank Wildlife Centre, near Dunfermline, before being released when it has fully recovered.
Animal welfare charities have to deal with an increased number of incidents of seals being washed ashore at this time of year because of the bad weather.
It is also the time of year when the grey seals are having their pups.
John Robins, secretary of the Save Our Seals Fund, said the baby seal at Portobello could have come from the other side of the Forth, or from elsewhere on the east coast, depending on the tide and weather.
He said the pup will probably have to spend two to three months in the rescue centre before it is ready to fend for itself in the wild.
Mr Robins said: "Because of the bad weather, there have been quite a few seals washed away from their mothers and we have to take them in and hand rear them.
"It will have to be taken into Middlebank and they will need to feed it with seal milk substitute, then on liquidised fish and get its weight up to three or four times what it is now.
"It will be two or three months before it is ready to go back to the wild.
"They will have to make sure it can fend for itself before it is released."
He added: "Members of the public who see one should not go too close and make sure they keep their dogs away.
"They should watch from a distance and see if the mother seal comes back. Sometimes she may be out fishing and will return with the food.
"If she doesn't come back between tides, then that's when they should call for help."
A Scottish SPCA spokeswoman said the seal pup, which was not injured, is likely to be taken to the Middlebank centre today. She added: "It wasn't abandoned because it is weaned and has lost all its white coat.
"It wasn't doing too well on its own.
"It was a bit under-weight, but it will get fed up and will be released at an appropriate time.
"We are starting to get more of these calls because of the poorer weather."