Hamas said on Saturday morning that its operatives were exchanging fire with IDF forces within Israeli lines, and claimed to have caused injuries.
"Our fighters are operating behind enemy lines. There are injuries on the other side," Hamas claimed.
The IDF would not confirm the reports.
Public Security Minister Yitzhak Aharonovitch said during a visit to Ashdod that there were clashes between IDF soldiers and terrorists on the Gaza border.
Sniper fire in the northern Gaza Strip wounded an IDF soldier on Saturday morning, the military confirmed.
Meanwhile, the Home Front Command prohibited travel in the Eshkol Regional Council due to the security situation.
The IDF on Saturday pressed ahead with a ground operation in the Gaza Strip, as Palestinian militants kept firing rockets deep into Israel's heartland.
Tanks were positioned in the morning near the border with Gaza, on the third day of the land campaign.
A rocket launched from Gaza landed in an open area near Kiryat Malachi mid-morning Saturday as a fresh volley of projectiles were launched at Israel. No injuries were reported.
Air raid sirens sounded intermittently in southern Israeli communities on Saturday morning, including in the Ashkelon Coast Regional Council, Ashdod, Gedera, Kiryat Malachi and Dimona.
Three IDF soldiers were wounded overnight during a confrontation with terrorists in the Gaza Strip. On Saturday morning the military confirmed that one soldier was seriously wounded and two others were lightly wounded in the incident.
The wounded soldiers were evacuated to an Israeli hospital for medical treatment.
US Secretary of State John Kerry was expected to arrive in Cairo on Saturday to discuss efforts to bring about a cease-fire between Israel and Hamas as the IDF's ground incursion launched Thursday night against the terror group continued, Egyptian media reported.
Channel 2's live footage filmed from outside the Gaza Strip intermittently showed isolated billows of smoke streaming out of from apparent explosions in Gaza.
Since the start of the IDF ground incursion, the army has struck 260 terrorist targets, 140 of which have been hit since Friday morning.
The IDF said it killed over 40 terrorists and arrested 21 others since the ground incursion started.