The two incidents in which international school students claimed they were being lured into vehicles were not cases of attempted kidnapping but instances of people trying to be helpful by offering the students a lift.
The police yesterday gave updates on the two cases, even as two other international schools this week sent out alerts over similar incidents.
In a statement, the police said they have thoroughly investigated the two cases reported on Jan 11 and 16, involving students from United World College (UWC) of South East Asia and Tanglin Trust School, respectively.
The reported incidents had sparked fears of kidnapping attempts - which the police have now ruled out.
In the first case on Jan 11, a male van driver had offered a ride to a female student from UWC as it was raining that day, said the police.
"The parents of the student had been updated on the findings, and they were relieved that it was a case of misunderstanding," they added.
In the second case on Jan 16, a female bus attendant on a school bus from Tanglin Trust School had noticed the student wearing its uniform walking towards the school.
"As the school bus was going to the school, the female bus attendant offered the student a ride. The school bus, however, did not bear the name or logo of the school," said the police. "The student declined the offer as she had earlier read a school circular advising them to be wary of strangers offering rides to students."
At Tanglin Trust School, drivers have to carry special passes identifying themselves as parents or guardians to enter the compound.
The police said they have clarified with the student that no person had alighted from the school bus to persuade her to board it.
"The police have established these two unrelated reports were not cases of attempted kidnapping. In both cases, the occupants of the vehicles were trying to be helpful by offering a lift to the students," they added.
The police also referred to another incident involving a student from Dulwich College in Bukit Batok reported by The Straits Times yesterday.
They said they have engaged the college to advise the student to make a police report.
In a letter sent yesterday morning to parents, Dulwich headmaster Nick Magnus said a senior school student heading home was approached in the vicinity of Farrer Road last month. The student, understood to be a boy, declined the offer of a lift and continued on his journey home, arriving safely.
The boy is understood to have revealed the incident after reports of two other international students who were offered lifts by strangers.
The police are also investigating another incident reported on Wednesday by the Nexus International School, located in Ulu Pandan.
In a letter sent to parents, the school said a female student was approached by two men in a vehicle at Old Holland Road on her walk home from school on Monday. It said the men were "quite persistent" despite her refusal to get into the car.
The police said they treat such reports seriously. "However, we urge members of the public not to speculate or spread unsubstantiated information which may generate unnecessary public alarm," the police statement said.
The incidents have opened up a debate about whether people offering lifts to children should be aware that their actions, although well meaning, will alarm children.
Several international schools, along with Fairfield Methodist Primary School, which is close to UWC in Dover Road, have sent out letters to parents urging them to be vigilant and take precautions.
• Additional reporting by Rachel Au-Yong