On Tuesday, President Andrzej Duda signed the controversial bill into law, which also bans people from referring to concentration camps as "Polish death camps".
Israel has said the law would curb free speech, criminalise basic historical facts and stop any discussion on the role that some Poles played in Nazi crimes. The law takes effect 14 days after it's officially published, but it wasn't immediately clear when that will be.
"I have chose to sign the law but also to send it to the Constitutional Tribunal", Duda told reporters in Warsaw.
That language seemed more conciliatory than earlier statements, suggesting the sides are seeking compromise.
It has caused a diplomatic crisis with Israel, which fears it will enable Poland to whitewash the role of Poles during the Holocaust.
The department warned that if the legislation is signed, it could have repercussions for "Poland's strategic interests and relationships".
The United States is "disappointed" that Poland's president has signed legislation that would impose criminal penalties for attributing Nazi crimes to the Polish state, said U.S. Secretary of State Rex Tillerson on Tuesday.
The EU has stayed quiet on the Holocaust law.
Polish officials have emphasized that artistic and historical research work will not be affected.
The law sets fines or a maximum three-year jail term for anyone who erroneously describes Nazi German death camps like Auschwitz-Birkenau as being Polish, simply due to their geographical location.
Duda said in a televised address that the law would protect the country's worldwide reputation. "[We] do not deny that there were cases of huge wickedness" in which Poles denounced Jews, he said, according to the AP.
It also criminalises denial of crimes against Poles by Ukrainian collaborators with Nazi Germany.
The new legislation has already sparked an uproar in Israel.
Ukraine, Poland's main ally in the former Soviet region, also complained, with MPs tabling a resolution that said they were "deeply concerned" the law would "strengthen anti-Ukrainian tendencies" in Polish society.
Jerzy Czerwinski, a senator with the ruling party, said on state radio Monday that he saw a "hidden agenda" in the opposition.
The bill first was proposed about two years ago, soon after Law and Justice took power in 2015, but hadn't been an issue of public debate recently.
Outrage was particularly loud in Israel after the bill passed through Poland's parliament on January 26, the day before Holocaust Remembrance Day, which many viewed as insensitive.