Proceedings against the Uruguay international began on Wednesday in front of an independent three-man panel but, although more aspects of the case were heard on Thursday, Press Association Sport understands no decision is imminent.
The 24-year-old has been charged with abusing Manchester United's Patrice Evra, and referring to his "ethnic origin and/or colour and/or race" by the Football Association.
Suarez has always denied the allegations levelled against him by the France defender and the player and his club were hoping to successfully make a case for his defence at the two-day hearing.
Manager Kenny Dalglish has given the Uruguay international his full support since the complaint was made back in October. And he did not waver from that stance when asked about the case at his pre-match press conference ahead of Sunday's Barclays Premier League match at Aston Villa.
"We are standing right beside him and helping him as much as we possibly can," said the Scot, who would not comment on reports he planned to appear in person at Suarez's hearing. "We have said many times before we cannot say anything which will be prejudicial. After it's finished there is a chance to have a conversation.
"As I said last week the FA should have given you (the media) the information when everything was happening, not us."
Evra made his allegations about Suarez immediately after the match at Anfield on October 15, telling French TV station Canal Plus the South American had used a racist insult to him "at least 10 times".
If Suarez, who has always denied the allegations, is found guilty he could be banned or fined, or both. Evra said: "There are cameras. You can see him (Suarez) say a certain word to me at least 10 times."
Both players spoke to FA officials during the investigation and there have been suggestions Suarez was unaware language that is acceptable in Uruguay is viewed as racist in Europe.