On Wednesday, Turkish forces helped Syrian rebels take the border town of Jarablus from so-called Islamic State.
Turkish Defence Minister Fikri Isik said on Thursday the operation had two goals – to secure the border area and ensure the Kurds “are not there”.
Ankara fears Kurdish gains in Syria will fuel an insurgency at home.
As Mr. Isik spoke, a column of some 10 Turkish tanks and a similar number of armoured vehicles reportedly crossed the border near Jarablus.
It was not immediately clear if the deployment was aimed at securing the town or helping members of the rebel Free Syrian Army push further into IS-held territory.
Rebel commanders said they had advanced up to 10km (6 miles) south of Jarablus and 10km westwards along the Turkish frontier.
The UK-based Syrian Observatory for Human Rights said fighters from the US-backed Syrian Democratic Forces (SDF), an alliance dominated by the Kurdish Popular Protection Units (YPG) militia, had meanwhile moved north from positions near Manbij, which they recently captured from IS.
A rebel source told the Reuters news agency that the two sides clashed at the village of al-Amarna, on the western bank of the Euphrates, on Wednesday night.
During a visit to Ankara on Wednesday, US Vice-President Joe Biden said Washington had warned the SDF not to move west of the Euphrates or risk losing American support.
Mr. Isik told NTV television: “If this withdrawal doesn’t happen, Turkey has every right to intervene.”
Earlier, a spokesman for the US-led coalition against IS tweeted that SDF fighters had moved east across the Euphrates from Manbij to prepare for “the eventual liberation” of the IS stronghold of Raqqa.