Prince Buaben, a Ghana-born player, was the target of the alleged behaviour.
The football club said it was ''hugely disappointed'' after a fan was ejected from Firhill stadium.
Police Scotland said a 42-year-old man was arrested in connection with the Offensive Behaviour at Football and Threatening Communications Act following the game against Hibernian.
He was expected to appear at Glasgow Sheriff Court, but the Crown Office said a report is still being considered by the procurator fiscal.
A spokesman said: "The procurator fiscal at Glasgow has received a report regarding an incident involving a 42-year-old man that allegedly occurred in Glasgow on 15 March.
"The report remains under consideration of the procurator fiscal".
Partick Thistle FC said in an earlier statement: ''Partick Thistle does not accept racism, sectarianism, homophobia or bigotry of any kind and anybody found to be using such language will be ejected and banned from attending games in the stadium. There is no place for bigotry at Firhill."
Thistle won the Scottish Premiership match 3-1.