Ms Goodwin said she was “cross” at the time, but did not report the incident.
No 10 said the Cabinet Office would look into any formal complaint made.
“Allegations such as this are taken very seriously,” the Downing Street spokesman added.
Ms Goodwin said the official – who has not been named – invited her into an office at Number 10 for the meeting.
She said she was surprised when the man, who was a few years younger than her, put his feet up on her chair and remarked that her sunglasses made her “look like a Bond Girl”.
She said she tried to steer the conversation back onto professional matters, but added: “At the end of the meeting we both stood up and the official, to my astonishment, put his hand on my breast.
“I looked at the hand and then in my best Lady Bracknell voice said: ‘Are you actually touching my breast?’
“He dropped his hand and laughed nervously.”
Ms Goodwin said she left Downing Street in a state of “high dudgeon”.
“I wasn’t traumatised, I was cross. But by the next day it had become an anecdote, The Day I Was Groped In Number 10,” she said.
A spokesman for Mr Cameron – who was in office from 2010 to 2016 – said he was first made aware of this “serious allegation” on Monday.
“He was alarmed, shocked and concerned to learn of it and immediately informed the Cabinet Office,” the spokesman added.
‘Keep Calm’ philosophy
Theresa May’s official spokesman said: “Of course this is something that we would be concerned about.
“We are looking at it, and as we have said, wherever an allegation has been made we will make sure it’s treated with the utmost seriousness.”
Ms Goodwin said recent revelations of alleged abuses had made her question whether she was wrong not to have made a formal complaint.
“Now, in the light of all the really shocking stories that have come out about abusive behaviour by men in power from Hollywood to Westminster, I wonder if my Keep Calm and Carry on philosophy, inherited from my parents, was correct?
“The answer is, I am not sure.”
Hollywood has been rocked by allegations against film mogul Harvey Weinstein and others.
At Westminster, several Conservative and Labour MPs are being investigated over claims of sexual misconduct.