It seems almost absurd. The President tweets that he is being investigated, and then his lawyer goes on TV and says that he is not being investigated. The message was somewhat muddled since it was unclear whether the lawyer was denying the existence of an investigation altogether, or making the more limited claim that: "The president has not been notified by anyone that he’s under investigation." As to the first, it is doubtful that the lawyer knows. As to the second, it is likely an accurate statement.
To see that consider how the US Attorney's Manual puts it (the USAM is the internal operating bible for federal criminal investigations -- essentailly the policy roadmap). The question we want to answer is whether, under standard policy, we might have expected that President Trump would be notified if he were under investigation. The USAM seems to make clear that the answer to that question is "no" or at least "not normally."
To see why this is so, begin with a distinction. The USAM defines three classes of people -- those who are mere "witnesses" to an event, and those who are either the "target" or "subject" of an ongoing criminal investigation. Here is how USAM section 9-11.151 puts the difference:
A "target" is a person as to whom the prosecutor or the grand jury has substantial evidence linking him or her to the commission of a crime and who, in the judgment of the prosecutor, is a putative defendant. An officer or employee of an organization which is a target is not automatically considered a target even if such officer's or employee's conduct contributed to the commission of the crime by the target organization. The same lack of automatic target status holds true for organizations which employ, or employed, an officer or employee who is a target.
A "subject" of an investigation is a person whose conduct is within the scope of the grand jury's investigation.
Given the early nature of the investigation, it would be horribly premature for the Special Counsel to have concluded that the President is the target of an investigation. He may well be a "subject" of the ongoing obstruction investigation (and perhaps other investigations as well) but there is nothing that I can see in the public record that would support the conclusion that the Special Counsel now considers him a "target." Indeed, that is the most natural conclusion of how we should read the colloquial language that the President is "under investigation." It is a statement that his conduct is the subject of inquiry and within the scope of an ongoing review -- conclusions that, in light of public evidence, seem indisputable.
What then of the idea that the President has received no "notice" that he is under investigation. Here the USAM is clear by inference but not directly. There is no provision in the manual that either requires or prohibits a prosecutor from notifying a subject of his status with regard to an investigation. What the USAM does say with regard to targets is that notification is discretionary and not mandatory, but it is encouraged. Here is the langauge of USAM section 9-11.153:
When a target is not called to testify pursuant to USAM 9-11.150, and does not request to testify on his or her own motion (see USAM 9-11.152), the prosecutor, in appropriate cases, is encouraged to notify such person a reasonable time before seeking an indictment in order to afford him or her an opportunity to testify before the grand jury, subject to the conditions set forth in USAM 9-11.152. Notification would not be appropriate in routine clear cases or when such action might jeopardize the investigation or prosecution because of the likelihood of flight, destruction or fabrication of evidence, endangerment of other witnesses, undue delay or otherwise would be inconsistent with the ends of justice.
And so, for the President, the fact that he has not received notice of an investigation is actually good news, in an odd way. When and if the Special Counsel sends him such notification it will likely reflect that the Counsel has concluded that the President is the target of an investigation -- that is that, in the words of the USAM, there is "substantial evidence" he has committed a crime.
And as for the claim that no investigation exists at all -- unless the President's lawyer can read minds, he really would have no way of knowing. The rest, as they say, is mere speculation.