Both sides have been trading blame for breaching a ceasefire pact
Israel has allowed fuel into the Gaza Strip, but kept border crossings closed to all other goods after a Palestinian rocket fire violated a truce.
This is the third consecutive day that border crossings into Gaza have been closed, preventing humanitarian and commercial goods into the area.
Israel Radio reported that two mortar rounds landed in an open field on the Israeli side of the border on Friday.
The military confirmed there had been two explosions, but not the cause.
Palestinian militants say they are a response to Israeli violations of the truce.
Prior to Friday, four rockets have been fired into Israel by Palestinian militants since an truce was agreed on 19 June.
An estimated 600,000 litres of industrial fuel will be allowed to pass through the Nahal Oz border terminal for Gaza's only power station, a European Union Official told the Reuters news agency. This will keep the plant running for several days.
The EU funds the deliveries of fuel to the power station.
Israeli officials confirmed the fuel delivery but said other border crossings remained closed.
On Thursday, the militant al-Aqsa Martyrs Brigades, linked to the Fatah faction of Palestinian Authority President Mahmoud Abbas, confirmed they fired a rocket, which came to ground without causing injury.
Other attacks earlier this week have been carried out by Islamic Jihad militants as retaliation they say for an Israeli raid in the West Bank, although the truce applies only to Gaza.
Gaza's Hamas rulers have called on all groups to respect the ceasefire but say they will not be Israel's "police".
Palestinian militants also accuse the Israelis army of shooting at farmers and fishermen and say Israel has been closing the cargo crossings into Gaza in violation of the ceasefire terms.
Israel says it will keep the crossings into Gaza closed while rocket attacks continue.
On Thursday Israeli and Egyptian officials were holding talks about Gaza in Cairo.
Israel, represented by envoy Ofer Dekel, wants to renew indirect talks on the release of Israeli soldier Gilad Shalit held in Gaza since the summer of 2006.
Hamas wants Israel to release hundreds of Palestinian prisoners in exchange for freeing Cpl Shalit.
However, Israel has previously rejected many of the names Hamas put forward.