The fire came a day after plans to oust Kofa as speaker sparked intense protest.
Several protesters, including an aide to former president George Weah, were arrested during Tuesday’s protest.
Rep. Fauco, a prominent figure in the House of Representatives, allegedly uploaded a video on Facebook in which he said: “If they want us to burn the chambers, we will burn them.”
Police Chief Coleman said: “Representative Foucault threatened to burn down the shared cells and it was burnt down a day after that. Therefore, he needs to clarify his statement.”
Coleman said Coffa also has to explain a Facebook post he made during the protests.
Neither Foucault nor Coffa responded publicly to Coleman’s comments.
The government has offered a $5,000 (£3,900) reward for more information about the fire.
President Joseph Boakai expressed his disappointment over the incident and ordered an investigation by the security services.
Liberia’s House of Representatives has been engulfed in a power struggle in which a faction of lawmakers claims to have removed and replaced Speaker Kofu.
Another group opposed the action, calling it unconstitutional.
The Supreme Court ruling did not resolve the internal dispute.
