Lord Ahmed was sentenced in 2009 after pleading guilty to dangerous driving, having sent and received text messages in the build up to a fatal crash on the M1 near Sheffield on Christmas Day in 2007.
The final message was sent two minutes before his Jaguar struck the stationary car of 28-year-old Slovakian Martyn Gombar.
No causal link was made between the text messages and Mr Gombar's death, though.
The Times said it gained four separate Urdu-English translations to verify Lord Ahmed's comments during a television interview, which it said were thought to have been made in April last year during his visit to Pakistan. The newspaper claims the peer is seen saying:
I was charged with a traffic offence, for texting while driving. A traffic offence, dangerous driving, and it should have gone to the magistrates court.
My case became more critical because I went to Gaza to support Palestinians. My Jewish friends who own newspapers and TV channels opposed this.