The Bramleys spent most of the 17 weeks they were missing with the two children in the Tralee area of Co Kerry, a spokesman for Cambridgeshire Constabulary told The Irish Times last night.
In that period, the spokesman added, there were two or three sightings of them in the Republic, and the Cambridgeshire police asked the Garda Siochana to investigate these.
The spokesman said gardai had interviewed the people who reported the sightings and believed they may have been genuine.
The Bramleys said in a statement yesterday that reported sightings of them in Britain during the period they were missing were mistaken.
RTE reported last night that the couple and the two children they are seeking to adopt spent some time in a mobile home in Fenit, about eight miles from Tralee. They stayed for three weeks in a seaside caravan owned by local businessman, Mr John Deady, who recalled a "nice, respectable couple".
"We had not an idea who they were, and only realised it was the missing English couple after they had gone, when we saw their pictures, and those of the children in the newspapers. "We told the Gardai and they said the information would be passed on to Dublin."
"We regarded them as ordinary English holidaymakers. We get a lot of people coming here at all times of the year for a break."
He added: "The two girls were always very well behaved, and it was evident that they very much loved those people we assumed to be their parents."
The Bramleys checked in at 4 p.m. on Saturday afternoon for a Ryanair flight from Farranfore airport, about 25 miles from Fenit, to Stansted Airport.
They had a meal in the airport restaurant and were taken on the flight before the other passengers.
RTE reported that the Bramleys appeared to be very protective of the two children during the hour-long flight to London.
They disembarked before the other passengers.
The spokesman for Cambridge Constabulary said that the Bramleys had no friends or relatives in Co Kerry and, so far as he knew, had never been there before.
Mr Bob Piarson, a spokesman for Cambridgeshire County Council described all at Cambridgeshire Social Services as "very relieved that we've got the children back safe. It's gone on far too long".