I've TUSB3210 MCU. It loads the program from serial EEPROM at startupI've tred to implement in-circuit programming of this EEPROM .

With SCL as a Push pull a few things becomes impossible:
1) The device can only operate as a master.
2) Any device MUST respond in the allotted time because this violates the IIC standard of "if you can not make it in time, just hold the clock till you are ready".

so ..
1) Your EEPROM must be of a type that always respond in time.
2) The download must take place with the TUSB as the master.

Based on this, and the fact that the infosheet states "download from IIC" your approach must be wrong, have a read of datasheets and appnotes. Also verify that your EEPROM is specified as "always responding immediately", some EEPROMS may kick in a wait now and then and will thus be unusable for this.

Erik