The plan - confirmed by a congressional aide to CBS, the BBC's US partner - must still be approved by lawmakers.
Last week President Biden said he would halt weapons shipments if Israel launched a major invasion of Rafah.
On Tuesday, Israeli tanks advanced deeper into residential areas the south-east of the Gazan city, Palestinians told Reuters.
Mr Biden said last week that he had delayed the shipment of 2,000lb (900kg) bombs to Israel because of how they might be used in such a ground operation.
Asked by CNN about the delay, Mr Biden said: "Civilians have been killed in Gaza as a consequence of those bombs."
Getty ImagesThe White House has told Congress it wants to send more than $1bn (£800m) in weapons to Israel, despite pressure over Israel's Rafah offensive.
The package would include tank rounds, mortars and armoured tactical vehicles, according to Reuters news agency
Republican lawmakers in Congress have introduced legislation intended to prevent any further pauses in weapons shipments to the US ally.
The House of Representatives this week will vote on a measure that requires the state department and defence department to ensure the "prompt delivery" of military equipment.
The White House has vowed to veto the bill, if it manages to pass the Senate, which is unlikely.
Israel launched a military campaign to destroy Hamas in response to the group's cross-border attack on southern Israel on 7 October, during which about 1,200 people were killed and 252 others were taken hostage.
More than 35,170 people have been killed in Gaza since then, including 82 in the past 24 hours, according to the territory's Hamas-run health ministry.
The package being sent to Congress is the first since Mr Biden's administration paused arms transfers to Israel last week. It would include $700m in tank ammunition, $500m in tactical vehicles and $60m in mortar rounds, according to the Associated Press news agency.
On Friday, a US state department report found that some American-made weapons provided to Israel may have been used in breach of international law.
While the report was a clear rebuke of some Israeli operations in Gaza, it stopped short of definitively saying that the Israel Defense Forces (IDF) campaign had breached international law.
It added that assurances it had received from Israel about adhering to the legal use of US weapons were "credible and reliable".
Last month, Congress passed a bill providing $95bn in aid to Israel, Ukraine and Taiwan.
Weapons transfers to Israel have become a political liability for Mr Biden ahead of this November's 2024 White House election.
News of the latest arms package emerged soon after voting ended in a presidential primary election in Maryland, where activists had urged voters to register a protest ballot against what they see as Mr Biden's favouritism towards Israel over the Gaza war.