If the Client is paying some GST under reverse charge for the services specified in sec.9(3). Can that GST be claimed in the same month ? Because our GST Site also is accepting the same and without payment of tax it is not allowing the credit to be taken.

Yes, ITC of GST under reverse charge can be claimed in the same month.
Rule 36 of CGST Rules specifies that input credit shall be availed on the basis of invoice issued under Section 31(3)(f), subject to payment of tax.
ITC can be availed subject to payment of tax, not after payment of tax
Hence ITC can be availed in the same month

Sir, you right but I have suggested for RCM availing ITC and to avoid litigation in future with department that If RCM Liability of a month has been paid within due date then avail & utilised ITC in same month and where is payment has been made after due then availment & utilisation of ITC in the month of Payment of RCM Liability.