Scholars of hadith are unanimously agreed that it is best to narrate a hadîth of the Prophet (peace be upon him) in the exact words in which it as been received.
As for narrating a hadith by conveying its meaning in other than the precise words in which it was received, this is a matter of long-standing scholarly disagreement.
Ibn Sîrîn, one of the leading scholars among the Successors, was opposed to doing so. This opinion has been attributed to a few Companions as well, but not with any certainty.
The majority of scholars – and the view deemed to be correct – is that it is permissible to relate a hadîth of the Prophet according to its meaning and not in its exact words, as long as the person who does so is knowledgeable about how the language conveys its meanings and is aware of what would alter the meaning of the text. Scholars agree that it is forbidden for a person to relate a hadîth by other than its exact words if that person does not have sufficient knowledge of the language to know the precise meanings of its words and what would change the text's original meaning.
It should be poined out that certain hadîth really should be related verbatim, without any changes whatsoever made to their wording. These are hadîth which provide words that we use as formula in our worship. These include the wording of the recitations we say in our prayers as well as prescribed supplications and remembrances. Since in these cases, the wording is significant, the narration of such hadîth demands an extra degree of vigilance.
As for the evidence showing the general permissibility of relating hadîth by their meanings and not in their exact wordings, we have the hadîth literature itself. If we consider the texts, we can see many cases where different Companions related their accounts of the exact same incident, wherein they varied in how they quoted the Prophet's statements. This indicates a degree of leniency when it comes to their adherence to the precise wording in what they narrated.
Someone might counter with the possibility that this variation in wording might have been introduced by later generations of narrators and not by the Companions themselves. However, there is no evidence to validate such a claim.
Moreover, we are not speaking about a few isolated hadîth. We are talking about hundreds of different hadîth that were narrated from the Prophet (peace be upon him). It is highly unlikely that all of this variation can be attributed to later generations of narrators. In a good number of cases, we can pinpoint that the disagreement in wording had to come from the different Companions who narrated the incident. This can be determined by examining how the various lines of transmission for the hadîth have reached us.
If we study the methods employed by the authors of the hadîth compilations, especially Sahîh al-Bukhârî and Sahîh Muslim, we can see that they were fully aware of this situation. These works are full of hadîth that relate the exact same circumstances with variant wordings for the Prophetic statements quoted therein. Examples include the hadith about Prophet's night journey, the hadîth about the sale of Jâbir's camel, and many other hadîth that come in the form of narratives of various events. In such cases, we know that we are dealing with a single event, so any variation in how the Prophet's words are quoted must be attributed to the narrators.
And Allah knows best.