Merseyside Police said they were called to the premises in Manor Road, Wallasey, on Tuesday morning following reports of criminal damage.
“It is believed that between 5pm last night and 9am today, damage was caused to the front ground-floor window of an office,” the force said in a statement.
“The incident was reported to Merseyside Police at 9.55am today. An investigation into the incident is ongoing.”
The incident follows the formal announcement on Monday by Ms Eagle that she is challenging Jeremy Corbyn for the Labour leadership.
Local councillor Bernie Mooney - a supporter of the MP - told the Liverpool Echo that she had just learned of the attack and was not sure of the damage.
“All I’ve been told is they have put the windows through. I just got the message and I’m going there now,” she said.
Ms Eagle may face moves by local party members to de-select her after the Wallasey Constituency Labour Party (CLP) had urged her to support Mr Corbyn.