The Speaker said the resignations of eight MLAs were not in the prescribed format and five others needed to explain why their action did not fall under the Anti-Defection law.
The legislators have been given time till July 21 to re-submit their resignations and explain reasons for quitting, he said.
Congress MLA Ramalinga Reddy, who has resigned, is said to be the key among the rebels and his daughter Soumya Reddy, also a Congress legislator, met UPA Chairperson Sonia Gandhi in Delhi on Tuesday, according to sources.
Of the 13 legislators who have tendered their resignations, 10 belong to the Congress and three to the JD-S.
The Speaker said in Bengaluru that on scrutiny, resignations of only five lawmakers were found to be in the proper format.
"As the remaining eight resignations of legislators of both the parties are not in the right format, I have given them time till July 21 to re-submit them and explain reasons for quitting from their respective Assembly constituencies," Kumar said.
"I have summoned all the legislators of both the parties (Congress and JD-S) to meet me on July 12, July 15 and July 21, as their resignations are faulty and explanations are not in accordance with the law under section 202 of the Anti-Defection Act," he said.
Of the five legislators, whose resignations were in order, Anand Singh, Ramalinga Reddy and Pratapgouda Patil are of the Congress while N. Narayana Gowda and K. Gopalaiah are of the JD-S.
"I have asked Singh, Patil and Gowda to meet me on July 12 to explain their resignations and ensure they were voluntary and genuine as per the law," Kumar said.
The Speaker summoned Reddy and Gopalaiah on July 15 to explain the reasons for their resignations.
The eight legislators whose resignations are faulty and have been told to resubmit them in the right format are B.C. Patil, S.T. Somshekar, Byrati Basavaraj, Ramesh Jarkiholi, Mahesh Kumathalli, Shivram Hebbar, Munirathna (all Congress) and A.H. Vishwanath of the JD-S.
On the resignation of suspended Congress legislator R. Roshan Baig, the Speaker said his (Baig's) letter was being verified by his office, as Baig submitted it on Tuesday afternoon.
The 13-month-old JD-S-Congress government has plunged into a crisis because of a spree of resignations, for which the ruling coalition blames the BJP.
In the 225-member state Assembly, the JD(S)-Congress alliance, along with a member of a regional party KPJP and an Independent, had the strength of 118, just five more than the required majority mark.
( With inputs from IANS )